/T7 


ABRIDGED  AND  APPLIED 
COLLEGE  PREPARATORY 


M.AUERBACH 

C.B.WALSH 


\ 


GIFT  OF 


PLANE  GEOMETRY 


LIPPINCOTT'S  SCHOOL  TEXT  SERIES 

EDITED  BY  WILLIAM   F.   R  USSELL,  PH.D. 

DEAN,     COLLEGE    OF     EDUCATION,    STATE    UNIVERSITY    OF   IOWA 


PLANE   GEOMETRY 

I.  ABRIDGED  AND  APPLIED 
II.  COLLEGE  PREPARATORY 


BY 
MATILDA  AUERBACH 

SUPERVISOR    OF    MATHEMATICS    IX    THE    ETHICAL    CULTURE    HIGH    SCHOOL,   NEW  YORK    CITY 

AND 
CHARLES  BURTON  WALSH 

PRINCIPAL    OF   THE    FRIENDS'    CENTRAL    SCHOOL,    PHILADELPHIA 


PHILADELPHIA,  LONDON,  CHICAGO 
J.   B.   LIPPINCOTT   COMPANY 


GA455 
A* 


COPYRIGHT,   1920,  BY  J.  B.  LIPPINCOTT  COMPANY 


PRINTED  BY  J.  B.  LIPPINCOTT  COMPANY 

AT  THE  WASHINGTON    SQUARE  PRESS 

PHILADELPHIA,  U.  S.  A. 


PREFACE 

IN  separating  this  text  book  on  Plane  Geometry  into  two  parts, 
the  Authors  have  followed  what  appears  to  them  to  be  a  normal 
and  logical  requirement  essential  to  a  proper  presentation  of  the 
subject,  and  the  most  appropriate  reference  to  these  divisions 
would  seem  to  be  a  designation  of  them  as  a  First  Study  and  a 
Second  Study.  In  the  former,  the  objective  is  to  afford  a  general 
view  of  the  subject,  with  emphasis  on  applications,  the  study  being 
intended  for  the  use  of  all  high  school  pupils,  and  the  material  is 
so  presented  as  to  make  it  available  also  for  use  in  junior  high 
schools.  The  Second  Study  is  devoted  to  a  more  intensive  treat- 
ment of  Plane  Geometry,  with  special  emphasis  on  theoretical 
work,  and  is  addressed  particularly  to  Regents7  and  college  entrance 
requirements.  In  the  entire  text  the  inductive  method  is  followed 
as  far  as  practicable,  and  simplicity  is  gained  rather  by  the  means 
of  scientific  accuracy  than  at  its  expense. 

In  view  of  the  fact  that  the  purpose  and  method  of  the  two  parts 
of  the  book  differ  somewhat,  a  separate  consideration  of  each 
of  these  divisions  seems  desirable. 

PART  I 

The  Authors  feel  confident  that  the  First  Study  will  serve  a 
fourfold  purpose. 

First:  That  it  will  contribute  to  a  solution  of  the  question  as  to 
how  much  mathematics  shall  be  required  of  high  school  pupils 
who  do  not  intend  to  enter  college,  and  with  this  objective,  an 
effort  has  been  made  to  plan  a  course  adapted  to  the  needs  and 
interests  of  pupils  meeting  minimum  requirements  in  mathematics. 
The  Authors  believe  that  it  has  been  customary  in  many  schools 
to  meet  this  situation  by  a  study  of  only  a  portion — three  or  four 
books  of  the  geometry  in  the  required  course,  thus  giving  students 
merely  an  intensive  knowledge  of  a  part  of  the  subject,  instead  of 
a  broadly  comprehensive  view.  The  course  here  outlined  covers 

459936 


vi  PREFACE 

not  only  all  that  is  really  important  in  the  five  books — although 
the  syllabus  contains  fewer  propositions  than  are  now  comprised 
in  the  first  three  books  of  most  available  texts — but  also  work  in 
the  application  of  three  of  the  trigonometric  functions. 

Second:  That  it  will  suggest  a  course  in  elementary  geometry 
so  thoroughly  adapted  to  the  mental  development  of  pupils  in  the 
ninth  or  tenth  school  year  that  it  may  be  profitably  used  to  super- 
sede the  conventional  course  in  formal  geometry. 

The  course  here  outlined  will  not  in  any  respect  restrict  the 
preparation  of  the  student  for  college;  on  the  contrary,  will  find 
him  much  more  ready  and  willing  to  proceed  to  the  collegiate 
preparatory  work  with  his  knowledge  of  the  subject  enriched  by 
application  and  vitalized  with  interest. 

Third:  That  it  will  open  the  eyes  of  the  pupil  to  the  relation  of 
geometry  to  the  activities  and  necessities  of  every-day  life,  and 
emphasize  the  practical  application  of  the  science,  both  in  specific 
professions  and  trades,  and  in  the  affairs  of  daily  life. 

Fourth:  That  it  will  arouse  in  the  pupil  a  conception  of  the 
dignity  and  power  of  the  subject.  To  this  end  the  Authors  have 
treated  it  scientifically,  endeavoring  to  develop  gradually  in  the 
mind  of  the  pupil  a  natural  assumption  of  this  treatment;  and  it 
•has  been  their  purpose,  by  departing  from  formal  methods,  to  lead 
the  pupil  to  reason  rather  than  merely  to  remember. 

The  following  means  have  served  in  the  attainment  of  the  ends 
just  stated: 

REVISION  OF  THE  SYLLABUS 

In  the  First  Study  the  NUMBER  OF  PROPOSITIONS  has  been 
reduced  to  approximately  half  of  those  given  in  the  standard  texts. 
Propositions  have  been  retained  or  selected  on  three  bases : 

First:     Those  that  are  rich  in  application. 

Second:  Those  of  peculiar  interest  to  the  young  student. 

Third:    Those  essential  to  the  sequence  of  the  study. 

The  wording  of  the  propositions  retained  has  departed  materially 
from  the  traditional  phraseology  in  an  effort  to  avoid  the 
formidable  and  stilted  qualities  of  the  latter,  while  retaining 
its  scientific  correctness.  In  fact,  the  language  of  the  students  hi 
the  classroom  has  suggested  many  of  these  changes:  e.g.  "Three 


PREFACE  vii 

sides  determine  a  triangle  "  to  replace  "Two  triangles  are  congruent 
if  three  sides  of  one  are  respectively  equal  to  three  sides  of  the 
other."  The  Authors  feel  that  some  of  these  changes  in  wording 
are  desirable  scientifically  as  well  as  from  a  practical  standpoint: 
e.g.,  "If  the  ratio  of  the  sides  of  one  triangle  to  those  of  another  is 
constant,  the  triangles  are  similar,"  to  replace  "Two  triangles  are 
similar  if  the  sides  of  one  are  respectively  proportional  to  the  sides 
of  the  other;"  and  "An  angle  whose  vertex  is  outside  a  circle  is 
measured  by  half  the  difference  of  the  intercepted  arcs,"  to  replace 
three  statements. 

A  transition,  more  gradual  than  usual,  from  the  geometry  of  the 
grade  school  to  the  more  scientific  work  of  the  secondary  school  has 
been  secured.  The  proofs  given  hi  the  first  section  would  be  wholly 
convincing  but  the  forms  and  detail  less  conventional  than  those 
demanded  in  more  rigorous  demonstrations,  some  of  which  are 
preceded  by  explanations.  This  does  not  mean  that  the  text  is  one  of 
Concrete  or  Observational  Geometry.  The  bare  essentials  are  retained 
from  the  outset,  and  subsequently  there  is  a  gradual  introduction 
and  demand  for  all  the  rigor  obtainable  in  secondary  school  work. 
This  gradual  transition  tends  to  prevent  the  discouragement  so 
often  manifested  during  the  beginning  of  the  study. 

MINOR  DETAILS. 

First:  No  authorities  are  required  for  auxiliary  construction — con- 
sistent reference  to  them  makes  a  proof  unduly  formidable — no 
other  reference  should  be  permitted. 

Second:  Nothing  is  introduced  in  the  text  until  it  is  required — thus 
avoiding  long  lists  of  definitions,  axioms,  postulates,  and  the  like. 

Third:  Throughout  the  text  emphasis  is  laid  upon  the  idea  of 
classification;  and  by  means  of  proper  grouping  of  definitions, 
postulates,  and  developed  facts,  the  student  is  trained  to  regard 
the  subject,  not  as  a  miscellany  of  isolated  facts,  but  as  a  frame- 
work of  interrelated  sub-topics.  A  few  reference  books  are  men- 
tioned. The  use  of  these  has  been  found  so  exceedingly  helpful 
and  inspiring  in  the  classes  which  have  been  led  by  the  Authors 
that  they  feel  that  the  omission  of  some  such  lists  would  be  a  great 
calamity.  The  Authors  recall  many  instances  in  which  new  life 


viii  PREFACE 

has  been  given  to  the  subject  through  such  references;  in  fact  there 
are  instances  where  a  pupil  who  might  never  have  discovered  him- 
self mathematically  has  developed  true  mathematical  enthusiasm 
and  ability  by  browsing  through  suggested  supplementary  reading. 

In  using  the  text,  the  Authors  earnestly  suggest  that  actual 
writing  of  proofs  be  deferred  until  the  class  is  quite  ready  to  fall  in 
naturally  with  a  more  or  less  set  form  of  proof. 

Let  the  need  of  a  form  become  apparent  to  the  student  who  is 
trying  to  write  a  proof  unaided  by  conventions  before  insisting 
upon  the  adoption  of  one  in  written  work.  Indeed  the  Authors 
feel  that  the  entire  first  chapter  of  this  text  could  well  be  developed 
before  the  necessity  arises. for  a  single  written  proof  from  the  stu- 
dent. Sufficient  material  for  written  work  will  be  available  from 
the  exercises  during  this  period. 

PART  II 

In  addition  to  a  review  of  the  First  Study,  the  Authors  desire 
to  direct  attention  to  some  details  of  this  Second  Division  of  the 
work  which  seem  worthy  of  special  mention. 

FIRST:  The  size  of  the  syllabus.  The  number  of  propositions 
developed  in  the  First  Study  has  been  considerably  augmented  to 
include  all  the  demands  of  College  Entrance  and  Regents'  Examina- 
tions. This  has  been  accomplished  by  inserting  the  additional 
theorems  between  the  theorems  of  the  original  syllabus,  thus 
preserving  the  sequence  of  the  former  Study.  A  separate  syllabus 
of  construction  problems  is  given  in  the  chapter  entitled  "  Methods 
of  Attacking  Problems  "  (page  306). 

SECOND  :  The  grouping  of  the  syllabus.  To  facilitate  the  reten- 
tion of  the  frame-work  of  the  subject,  the  propositions  are  collected 
in  groups  by  topics,  so  far  as  the  sequence  permits,  and  such 
grouping  has  necessitated  certain  departures  from  the  traditional 
arrangement  by  books. 

THIRD  :  The  type  of  exercises.  In  this  part  emphasis  is  placed 
on  theoretical  exercises,  as  contrasted  with  special  reference  to  prac- 
tical exercises  in  the  First  Study;  and  a  large  collection  of  college 
entrance  papers,  together  with  a  still  larger  selection  of  isolated 
exercises  of  this  character,  is  included. 


PREFACE  ix 

FOURTH  :  A  chapter  on  methods  of  proof.  More  scientific  habits 
of  work  are  fostered  through  a  discussion  and  a  careful  classification 
of  methods  of  proof  with  illustrative  exercises. 

FIFTH  :  A  chapter  on  methods  of  attacking  probkms.  This  chapter 
is  largely  similar  in  purpose  to  the  chapter  just  mentioned,  except 
that  it  deals  with  construction  work.  A  special  syllabus  of  what 
might  be  termed  fundamental  constructions  is  given,  including 
those  presented  hi  standard  texts,  and  those  propositions  are  of 
such  character  as  to  form  a  necessary  and  sufficient  basis  for  the 
work  required  by  colleges.  This  chapter  groups  many  typical 
constructions  and  methods  employed,  and  definite  reference 
throughout  the  book  is  made  to  this,  as  well  as  to  the  chapter  on 
methods  of  proof. 

SIXTH  :  Suggestions  for  club  or  other  additional  work.  The  chap- 
ter entitled  "Suggestions,"  and  exercises  preceded  by  the  letter 
"d" — frequent  in  the  book — may  be  omitted  (not  constituting  a 
requirement  for  college  preparation)  without  impairing  the  integ- 
rity of  the  course.  They  are  included  to  give  additional  interest 
and  breadth  to  the  subject  where  time  and  the  ability  of  the  student 
permit.  This  chapter  contains  in  the  suggestions  for  club  work  a 
list  of  topics  suitable  for  discussion  by  students  or  teacher  in  a 
mathematics  club  of  high  school  grade,  and  appended  to  this  list 
will  be  found  a  bibliography  of  appropriate  references. 

In  summary,  then,  it  may  be  said  that  this  Second  Study  is 
intended  to  enlarge  upon  the  course  outlined  in  the  First  Study, 
not  only  in  that  it  answers  the  requirements  of  college  entrance 
examinations,  but  in  that  it  also  makes  possible  at  the  same  time 
a  richer  and  fuller  course  for  those  students  whose  interest  and 
ability  warrant  it. 

In  closing,  the  Authors  desire  to  acknowledge  three  distinct 
sources  of  assistance. 

Realizing  the  present  eclectic  tendency  of  teachers  in  the  matter 
of  exercises,  as  evidenced  by  the  general  use  of  typewritten  lists 
of  problems,  the  Authors  have  availed  themselves  frequently  of 
many  of  the  standard  texts  in  the  selection  of  material  of  this  kind. 
To  Mr.  Lewi  Tonks,  a  former  pupil  of  the  Authors,  who  read  the 
proof  and  criticised  the  contents  of  the  book,  the  Authors  fee) 


x  PREFACE 

especially  indebted,  and  wish  at  the  same  time  to  acknowledge  the 
assistance  of  Mr.  H.  W.  Smith,  of  the  Ethical  Culture  School,  and 
of  Mr.  P.  S.  Clarke,  of  Pratt  Institute,  Brooklyn,  whose  suggestions 
for  revision  of  the  English  of  the  text  were  of  value. 

Finally,  the  Authors  take  pleasure  in  this  opportunity  to  record 
their  deep  appreciation  of  the  endorsement  and  encouragement 
they  have  received  from  the  authorities  of  the  Ethical  Culture 
School — Prof.  Felix  Adler,  Rector;  Franklin  C.  Lewis,  Superin- 
tendent; and  Dr.  Henry  A.  Kelly,  High  School  Principal.  Their 
approval  made  possible  the  experimental  work  in  the  School  which 
has  developed  and  justified  the  course  here  presented,  and  the 
Authors  feel  that  their  kindly  sympathy  and  intelligent  cooperation 
in  the  growth  of  the  experiment  have  contributed  in  large  measure 
to  its  success. 

THE  AUTHORS. 
OCTOBER,  1919. 


SYMBOLS  AND  ABBREVIATIONS 


=  is  equal,  or  equivalent,  to 

^  is  not  equal,  or  equivalent,  to 

«*  is  similar  to 

^  is  congruent  to 

=  approaches  as  a  limit 

oc  varies  as 

<*_  is  measured  by 

>  is  greater  than 

>  is  not  greater  than 
<  is  less  than 

<£  is  not  less  than 

+  plug;  or  increased  by 

minus,  or  diminished  by 

-T-  ,  —  divided  by 

X,()/  multiplied  by 

a  parallel,  or  is  parallel  to 

JT  not  a  parallel,  or  is  not  parallel  to 

||s  parallels 

J_  a  perpendicular,  or  is  perpendicu- 

lar to 

JL  not  a  perpendicular,  or  is  not  per- 

.    pendicular  to 

-Ls          perpendiculars 

arc 

A  B          straight  line  AB 
O  (D         circle,  circles 
A,  A        triangle,  triangles 
O,  EU        parallelogram,  parallelograms 
"£  ;  *£3         angle,  angles 
since 
therefore 

and  so  on 

>{}>[]'       signs  of  aggregation 
V  the  nth  root  of 

TT  3.14159.... 


adj. 

adjacent 

alt. 

alternate 

ap. 

apothem 

approx. 

approximately 

ax. 

axiom 

circf. 

circumference 

comp. 

complement,  com- 

plementary 

con. 

conclusion 

cong. 

congruent 

const. 

construction 

cor. 

corollary 

corres. 

corresponding 

def. 

definition 

diff. 

difference 

ex. 

exercise 

ext. 

exterior 

fig. 

figure 

ht. 

height,  or  altitude 

horn. 

homologous 

hy. 

hypotenuse 

hyp. 

hypothesis 

int. 

interior 

isos. 

isosceles 

lat. 

lateral 

peri. 

perimeter 

pi. 

plane 

pt.,  pts. 

point,  points 

n-gon 

polygon  of  n  sides 

post. 

postulate 

prob. 

problem 

proj. 

projection 

prop. 

proposition 

rect. 

rectangle 

reg. 

regular 

rt. 

right 

sec. 

sector 

seg. 

segment 

sq. 

square 

St. 

straight 

subst. 

substitute 

sup. 

supplement,    sup- 

plementary 

sym. 

symmetrical 

th. 

theorem 

vert. 

vertical 

si 

CONTENTS  PAGE 

PREFACE  v 

LIST  OF  SYMBOLS  AND  ABBREVIATIONS xi 

PART  ONE— FIRST  STUDY 

CHAPTER  I 

INTRODUCTION 

A.  A  FEW  FACTS  CONCERNING  THE  EARLY  DEVELOPMENT  OF  GEOMETRY  3 

Summary 10 

Bibliography 10 

Exercises.    Set  I.    Based  Upon  Historic  Facts 11 

B.  A  FEW  ILLUSTRATIONS  OF  GEOMETRIC  FORM 12 

Exercises.    Set  II.    Illustrations  of  Geometric  Form. .  .* 13 

C.  MEANING  OF  GEOMETRIC  FORM 14 

Exercises.    Set  III .    Meaning  of  Geometric  Form 15 

D.  SUGGESTIONS  OF  A  FEW  USES  OF  GEOMETRY 16 

Exercises.    Set  IV.    Mensuration 17 

Exercises.    Set  V.    Constructions.    Designing 17 

Exercises.    Set  VI.    Some  Other  Uses  of  Geometry 24 

E.  THE  BASIC  PRINCIPLES  OF  GEOMETRY 24 

Axioms 25 

Exercises.  Set  VII.  Illustrations  of  Postulates 26 

Postulates  of  the  Straight  Line 27 

Definitions 27 

Exercises.  Set  VIII.  Sums  of  Angles  in  Pairs 28 

Exercises.  Set  IX.  Measurement  of  Angles 29 

Postulates  of  the  Angle 31 

Exercises.  Set  X.  Relative  Position  of  Angles  31 

Exercises.  Set  XI.  Instruments  for  Measuring  Angles 36 

F.  THE  DISCOVERY  OF  SOME  FACTS  AND  THEIR  INFORMAL  PROOF  ....  36 

I.  Classification  Based  upon  Sides 37 

II.  Classification  Based  upon  Angles 37 

Experiments.     Theorems 37 

Exercises.    Set  XII.    Meaning  of  Congruence  and  Classi- 
fication of  Triangles 38 

Exercises.     Set   XIII.     Application   of   Congruence   of 

Triangles 39 

III.  Some  Properties  of  the  Isosceles  Triangle 42 

Experiment.    Theorem 42 

a.  Some  Properties  of  the  Equilateral  Triangle 43 

Exercises.    Set  XIV.    Equilateral  Triangles 43 

siii 


xiv  CONTENTS 

PAGE 

IV.  Further  Discussion  of  Congruence  of  Triangles 44 

Experiment.    Theorem 44 

Exercises.    Set  XV.    Further  Applications 44 

Summary  of  Chapter 47 

CHAPTER  II 

THE  PERPENDICULAR,  THE  RIGHT  TRIANGLE,  AND  PARALLELS 

A .  THE  PERPENDICULAR 48 

Postulate.  Theorem 48 

Exercises.  Set  XVI.  Distance  from  a  Point  to  a  Line 48 

Axioms  of  Inequality 49 

Exercises.  Set  XVII.  Numerical  Inequality 49 

Exercises.  Set  XVIII.  Inequality  of  Sects 50 

B.  THE  RIGHT  TRIANGLE 50 

Experiment      Theorems 50 

C.  PARALLELS 51 

Exercises.    Set  XIX.    Parallels 52 

Postulate  of  Parallels 52 

Exercises.    Set  XX.    Relative  Position  of  Angles .% .  53 

Exercises.    Set  XXI.    Construction  of  Parallels .' .  55 

Exercises.    Set  XXII.    Related  Statements 56 

Exercises.    Set  XXIII.    Applications  of  Parallelism 58 

Summary  of  Chapter 59 

CHAPTER  III 
ANGLES  OF  POLYGONS  AND  PROPERTIES  OF  PARALLELOGRAMS 

A.  ANGLES  OF  POLYGONS 60 

Exercises.    Set  XXIV.    Sum  of  Angles  of  a  Triangle 60 

Exercises.    Set  XXV.    Sums  of  Angles  01  Polygons 62 

Exercises.    Set  XXVI.    Sides  and  Angles  of  a  Triangle 66 

B.  PARALLELOGRAMS 67 

Exercises.    Set  XXVII.    Parallelograms 68 

Exercises.    Set  XXVIII.    Parallels 73 

Summary  of  Chapter 73 

CHAPTER  IV 

AREAS 

A.  INTRODUCTION.    REVIEW  OF  FRACTIONS 74 

Underlying  Principles 74 

Exercises.    Set  XXIX.    Fractions 75 

B.  AREAS.    DEVELOPMENT  OF  FORMULAS 77 

Exercises.    Set  XXX.    Comparison  of  Sects 78 

Exercises.    Set  XXXI.    Areas  of  Rectangles 80 


CONTENTS  xv 

PAGE 

Exercises.    Set  XXXII.    Areas  of  Parallelograms 81 

Exercises.    Set  XXXIII.    Altitudes  of  Triangles 81 

Exercises.    Set  XXXIV.    Areas  of  Triangles 82 

Exercises.    Set  XXXV.    Areas  of  Trapezoids 83 

Summary  of  Chapter 85 

CHAPTER  V 

ALGEBRA  AS  AN  INSTRUMENT  FOR  USE  IN  APPLIED  MATHEMATICS 

A.  LOGARITHMS 86 

I.  Introduction 86 

Exercises.    Set  XXXVI.    Meaning  of  Logarithms 86 

Exercises.    Set  XXXVII.    Applications  of  Laws  of  Expo- 
nents   88 

Exercises.    Set  XXXVIII.    Use  of  Tables  of  Powers 90 

Historical  Note 90 

II.  Principles  of  Common  Logarithms 91 

Exercises.    Set  XXXIX.    Common  Logarithms 92 

III.  Fundamental  Theorems 93 

IV.  Use  of  the  Table  of  Common  Logarithms 94 

Exercises.    Set  XL.    Use  of  Table 95 

Exercises.    Set  XLI.    Computation  by  Logarithms 98 

B.  RATIO,  PROPORTION,  VARIATION 105 

I.  Ratio  and  Proportion 105 

Exercises.    Set  XLII.    Ratio 106 

Exercises.    Set  XLIII.    Proportion 106 

Exercises.    Set  XLIV.    Applications  of  Proportion 108 

II.  Variation 110 

Exercises.    Set  XLV.    Applications  of  Variation Ill 

Summary  of  Chapter . . . 114 

CHAPTER  VI 
SIMILARITY 

A.  INTRODUCTORY  EXPERIMENTS  AND  THEOREMS 115 

Exercises.    Set  XLVI.    Proportional  Sects 116 

B.  IDEA  OP  SIMILARITY 119 

Exercises.    Set  XL VII.    Meaning  of  Similarity 121 

C.  SIMILARITY  OF  TRIANGLES 122 

Exercises.    Set  XL VIII.    Similarity  of  Triangles 124 

D.  PERIMETERS  AND  AREAS  OF  SIMILAR  TRIANGLES 132 

Exercises.    Set  XLIX.    Areas  of  Similar  Triangles 133 

E.  APPLICATIONS  OF  SIMILAR  TRIANGLES 133 

Exercises.    Set  L.    Projections.    Pythagorean  Relation 135 

Exercises.    Set  LI.    Trigonometric  Ratios 140 

Summary  of  Chapter 148 


xvi  CONTENTS 

CHAPTER  VII 

THE  Locus 

PAGE 

A.  REVIEW  OF  ALGEBRAIC  IDEA  OF  Locus 149 

Exercises.    Set  LII.    Location  of  Points 149 

Exercises.    Set  LIII.    Applied  Problems  in  Graphic  Mathematics  150 

Exercises.    Set  LIV.    Graphic  Solution  of  Equations 151 

Exercises.    Set  LV.    The  Equation  as  the  Statement  of  a  Locus  152 

B.  PECULIARITY  OF  THE  PROOF  OF  A  Locus  PROPOSITION 155 

Exercises.    Set  LVI.      Related    Statements — Direct,    Converse, 

Opposite 155 

Theorems 156 

Exercises.    Set  LVII.    Applications  of  Locus 157 

Summary  of  Chapter 159 

CHAPTER  VIII 

THE  CIRCLE 

DEFINITIONS 160 

A.  PRELIMINARY  THEOREMS 161 

Exercises.    Set  LVIII.  Circle  as  a  Locus 161 

B.  STRAIGHT  LINE  AND  CIRCLE 162 

Exercises.    Set  LIX.    Congruence  of  Curvilinear  Figures 163 

Exercises.    Set  LX.    Constructions  Based  Upon  Circles '. .  164 

Exercises.    Set  LXI.    Equal  Chords 165 

Exercises.    Set  LXII.    Tangent  and  Circle 167 

Exercises.    Set  LXIII.    Tangent  Circles 168 

C.  THE  ANGLE  AND  ITS  MEASUREMENT 172 

Exercises.    Set  LXIV.    Secant  and  Circle 173 

Exercises.    Set  LXV.    Circles 174 

Exercises.    Set  LXVI.    Inscribed  Angles 175 

Exercises.    Set  LXVII.    Measurement  of  Angles 177 

Exercises.    Set  LXVIII.    Tangent  and  Secant 179 

D.  MENSURATION  OF  THE  CIRCLE 180 

Exercises.    Set  LXIX.    Regular  Polygons  and  Circles 181 

Historical  Note 183 

Postulates  of  Limits 184 

Exercises.    Set  LXX.    Perimeters  of  Regular  Polygons 187 

Axioms  of  Variables 187 

Historical  Note 188 

Exercises.    Set  LXXI.    Value  of  TT 188 

Exercises.    Set  LXXII.    Circumference 189 

Exercises.    Set  LXXIII.    Area  of  Circle 192 

Summary  of  Chapter 195 

Miscellaneous  Exercises.    Set  LXXIV 197 


PART  I 

FIRST  STUDY 


• 


PLANE  GEOMETOY-   r  j 

CHAPTER  I 

INTRODUCTION 

NOTE. — For  pupils  who  have  had  no  intuitional  geometry 
section  A  of  the  Introduction  may  be  postponed  until  the  work  in 
Areas  has  been  completed — p.  77. 

A.  A  FEW  FACTS  CONCERNING  THE  EARLY  DEVELOP- 
MENT OF  GEOMETRY 

In  a  cabinet  in  the  British  Museum  there  is  a  piece  of  clay  some- 
what over  an  inch  thick  and  perhaps  fifteen  inches  square  which 
might  be  referred  to  as  the  first  book  about  geometry.  Near  it, 
on  a  roll  of  papyrus,  yellowed  by  age,  is  a  collection  of  notes  con- 
taining instructions  for  finding  the  contents  of  areas  and  solids. 
When  we  reflect  that  this  clay  tablet  and  the  manuscript  are  con- 
siderably over  thirty-five  hundred  years  old,  we  can  see  that  the 
study  of  geometry  is  by  no  means  a  modern  development. 

The  tablet  and  the  manuscript  represent  respectively  the  earliest 
available  records  of  the  geometric  knowledge  of  the  Babylonians 
and  the  Egyptians.  Centuries  ago  these  two  races  found  it  neces- 
sary to  devise  some  means  for  accomplishing  what  today  seems  a 
very  simple  undertaking.  Perhaps  the  necessity  was  forced  upon 
the  Egyptians  for  a  reason  that  does  not  seem  very  apparent  at 
first.  The  River  Nile,  as  we  know,  rises  twice  a  year  and  inundates 
the  country  bordering  on  it  for  many  miles.  Naturally  this  flood 
produces  changes  in  the  line  of  the  river  banks,  and  new  turns  and 
curves  give  the  adjacent  land  a  very  different  appearance  on  each 
occasion.  A  farmer  whose  land  bordered  the  river  might  therefore 
find  himself  one  year  in  possession  of  a  good  deal  of  property,  and 
the  next  year  with  much  less.  This  condition,  we  are  told  by  the 
historian  Herodotus,  caused  Ranleses  II,  who  was  king  of  Egypt 
about  1350  B.C.,  to  declare  a  law:  "This  king  divided  the  land 
among  all  Egyptians  so  as  to  give  each  a  quadrangle  of  equal  size, 

3 


4  PLANE  GEOMETRY 

«•    v «.  -  ;„  t   t    '  v 

and  to  draw^oln  each  his  revenues  by  imposing  a  tax  to  be  levied 

yearly;  but  everyone  from  whose  part  the  river  tore  away  anything 
hatl  to  go  ta  him  aiid  notify  him  of  what  had  happened;  he  then 
sent  the  overseers  who  had  to  measure  out  by  how  much  the  land 
had  become  smaller,  in  order  that  the  owner  might  pay  on  what 
was  left  in  proportion  to  the  entire  tax  imposed."  As  a  result  of 
this  it  became  necessary  for  the  Egyptians  to  employ  surveyors 
who  should  determine  the  areas  of  the  land  lost  or  gained,  and  these 
surveyors  put  into  practical  use  such  rudimentary  knowledge  as 
was  then  available. 


Restoration  of  the  Great  Hallof  Karnak 

Seven  centuries  before  this  Egyptian  kings  had  undertaken  large 
construction  operations,  the  very  nature  of  which  showed  that 
their  contractors  and  builders  understood  the  elementary  prin- 
ciples of  the  science  of  mensuration.  Menes,  the  first  Egyptian 
king,  built  a  large  reservoir  and  two  temples  at  Phthah  and  Mem- 
phis, the  ruins  of  which  are  still  in  existence,  and  under  Amenembat 
III,  a  later  king,  the  Egyptians  designed  and  constructed  a  very 
large  irrigating  system  covering  considerable  territory  and  requir- 
ing a  careful  calculation  of  areas,  water  flow,  gradients,  etc.  Stu- 
dents of  Egyptian  art  and  religion  find  frequent  evidence  that  this 
race  had  a  crude  knowledge  of  geometrical  principles.  The  pave- 
ments of  the  temples  show  designs  of  triangles,  squares,  five- 
pointed  stars,  and  rectangles,  and  the  locations  of  the  buildings 
themselves  show  geometric  knowledge,  for  their  temples  were 
supposed  to  be  constructed  with  reference  to  a  certain  fixed  point. 
The  Egyptians  were  sun-worshippers,  and  their  temples  were 


INTRODUCTION  5 

designed  to  receive  sunlight  through  the  doorways  at  certain  times 
of  the  day,  as  a  part  of  the  religious  ceremonies.  It  is  interesting 
to  note  that  the  movement  of  the  North  star  has  been  in  a  measure 
demonstrated  to  our  later-day  astronomers  by  the  fact  that  Egyp- 
tian temples,  built  three  or  four  thousand  years  ago  and  designed  to 
face  the  North  star,  are  no  longer  in  the  perpendicular  to  it.  The 
Egyptians  were  astronomers,  and  in  locating  their  temples  used 
the  sun  and  the  North  star  to  establish  base  lines.  The  surveyors, 
called  the  "  harpedonaptse "  or  " rope-stretchers," 
fixed  the  right  angle  to  the  north-south  line  by 
stretching  a  rope  knotted  in  three  places  around 
pegs.  The  distances  between  the  knots  were 
in  the  ratio  of  3-4-5,  showing  that  they  knew 
this  to  be  the  ratio  of  sides  of  a  right  triangle. 
Our  present  day  surveyors  are  still  following  the  same  method 
and  have  improved  upon  the  method  of  the  Egyptians  only  by 
substituting  a  steel  tape  for  the  rope. 

We  mentioned  the  ancient  manuscript  of  the  Egyptians  now  in 
the  British  Museum.  The  writer  of  this  manuscript  was  called 
Aah-mesu  (The  Moon-born),  an  Egyptian  scribe  commonly  called 
Ahmes.  The  original  from  which  he  copied  it  was  probably  in 
existence  about  2300  B.C.,  but  has  never  been  discovered.  The 
commercial  value  of  the  document  is  shown  by  the  fact  that  it 
contained  rules  and  formulas  for  finding  the  capacity  of  the  wheat 
warehouses  constructed  in  ancient  Egypt,  as  well  as  a  treatise  of 
considerable  length  on  a  crude  algebraic  system.  A  temple  built 
for  the  worship  of  the  god  Horus  on  the  island  of  Edfu  has  on  its 
walls  hieroglyphics  describing  the  land  which  the  priests  of  the 
temple  owned,  and  the  formulas  for  finding  the  areas  of  these  plots. 
Less  is  known  about  the  Babylonians  in  these  par- 
ticulars, but  so  far  as  we  can  learn,  their  geometrical 
knowledge  was  used  more  in  the  arts  than  for  practical 
purposes.  Their  monuments,  found  in  the  ruins  of 
Babylon,  show  geometrical  designs,  such  as  a  regular 
hexagon  in  a  circle,  and  the  pictures  of  their  chariots  show  the 
wheels  divided  into  sixths.  The  Babylonians  appear  to  have  fol- 
lowed this  division  into  sixths  in  their  arrangement  of  the  calendar, 


PLANE  GEOMETRY 


for  their  year  consisted  of  three  hundred  sixty  days,  and  they 
divided  the  circle  into  three  hundred  sixty  degrees,  on  the  theory 
that  each  degree  represented  the  supposed  revolution  of  the  sun 
round  the  earth. 

Although  the  geometric  knowledge  of  the  Egyptians  and  Baby- 
lonians may  seem  to  us  somewhat  crude  and  simple,  we  must 
remember  that,  as  compared  with  the  savage  races  which  sur- 
rounded them,  these  people  represented  the  greatest  advancement 
in  civilization  and  scientific  knowledge.  We  see  that  much  of  this 
was  due  to  the  very  necessities  of  life;  that  to  build  public  works, 
levy  taxes,  determine  boundaries,  required  a  knowledge  of  the 
science  of  mensuration.  In  the  case  of  the  Egyptians,  their  require- 
ments, so  far  as  we  are  able  to  estimate  them,  were  even  broader 
than  those  of  the  Babylonians.  The  construction  of  the  pyramids 

shows  clearly  a 
geometrical  design, 
executed  scientifi- 
cally, and  this  work, 
as  well  as  the  erec- 
tion of  wheat  ware- 
houses and  storage 
reservoirs,  necessi- 
tated what  was 
doubtless  to  them 
a  very  advanced  conception  of  the  principles  of  solid  geometry. 
Strangely  enough,  however,  we  are  indebted  to  neither  of  these 
races  for  the  development  of  this  knowledge  into  a  science,  but  to 
a  race  whose  place  in  history  is  much  later.  The  Greeks  in  the 
ancient  world  occupied  a  position  in  some  respects  similar  to  that 
which  America  has  held  in  the  modern  world.  They  were  a  people 
much  given  to  exploitation  and  expansion,  as  well  as  to  scientific 
and  philosophical  pursuits,  and  hi  addition  to  this,  prided  them- 
selves on  their  high  degree  of  adaptability.  Plato  said,  "Whatever 
we  Greeks  receive  we  improve  and  perfect."  They  did  not  origin- 
ate ideas  so  much  as  they  adopted  those  of  other  races  and  improved 
upon  them  to  a  degree  which  causes  history  to  associate  the  Greeks 
themselves  with  the  original  conception.  The  Greeks  were  travelers 


Egyptian  Pyramids 


INTRODUCTION 


and  traders,  interested  in  the  arts  and 
sciences,  and  a  distinguishing  character- 
istic of  the  race  was  their  desire  to  learn 
and  experiment  with  new  things.  Seven 
hundred  years  before  Christ,  Greek  mer- 
chants began  sending  their  ships  across 
the  Mediterranean  to  Egypt.  Travelers 
began  to  bring  back  accounts  of  this  other 
great  nation,  and  the  Greeks  were  imme- 
diately interested  in  the  reports  of  what 
the  Egyptians  had  done  and  were  doing. 

Thales  (640-546  B.C.),  a  merchant  of 
Miletus,  was  among  those  who  became 
acquainted  with  the  Egyptians  as  a  result 

of  commercial  intercourse.  Thales  was  at  heart  a  student,  and  the 
geometrical  theories  and  practices  of  the  Egyptians  interested 
him.  In  later  life  he  terminated  his  business  activities  and  opened 
a  school  in  his  native  city,  Miletus,  where  he  began  teaching  the 
principles  of  geometrical  science  as  it  was  then  known.  The 
problems  with  which  his  pupils  concerned  themselves  would 
seem  elementary  to  us.  They  dealt  merely  with  finding  the 
heights  of  objects  or  the  distances  of  ships  from  the  shore,  but 
his  school,  which  has  come  down  to  us  under  the  name  of  the 
Ionic  school  (so  called  from  the  Greek  province  in  which  Miletus 
was  situated),  was  the  first  intelligent  effort  to  systematize  the 
study  of  geometry. 

One  of  the  students  in  the  school  of  Thales  was  a  noteworthy 
successor  of  the  first  Greek  geometrician.  Pythagoras  (580-501 
B.C.)  founded  a  school  of  mathematics  at  Crotona  in  the  southern 
part  of  Italy.  His  plan  was  much  more  elaborate  than  that  of 
Thales.  Pythagoras  felt  that  the  study  and  character  of  the  school 
would  create  a  deeper  impression  if  it  were  organized  as  a  secret 
society.  The  watchword  was  " Silence,"  and  its  members  were 
pledged  to  secrecy  as  to  the  nature  of  the  work  which  was  done. 
The  Greek  government  felt  that  the  secret  methods  of  the  school 
might  be  used  to  conceal  harmful  activities,  and  finally  ordered 
the  institution  closed.  This  circumstance,  and  the  pledge  of  secrecy 


8 


PLANE  GEOMETRY 


imposed  upon  the  members  of  the  school,   have  prevented  our 
learning  much  about  it.  The  Pythagorean  proposition,  which  states 

that  the  square  on  the  hypothenuse  of 
a  right  triangle  is  equal  to  the  sum  of 
the  squares  on  the  other  two  sides,  bears 
the  name  of  the  school,  although  the 
fact  was  known  for  a  long  time  before 
Pythagoras  proved  it  to  be  true.  "  Py- 
thagoras changed  the  study  of  geometry 
into  the  form  of  a  liberal  education, 
for  he  examined  its  principles  to  the 
bottom  and  investigated  its  pro- 
Pythagoras  positions  in  an  immaterial  and  intel- 

lectual manner." 

Archytas  (430-365  B.C.),  who  followed  Pythagoras,  was  not  so 
much  interested  in  speculative  or  geometrical  subjects  as  he  was 
in  the  application  of  the  science  to  practical  uses.  He  invented 
several  mechanical  toys  operated  on  geometrical  principles.  Very 
few  sailors  realize  that  the  ability  of  one 
man  to  move  a  tremendous  weight  of 
sail  was  made  possible  by  the  discovery 
of  this  Greek  mathematician  who  lived 
over  twenty  centuries  ago,  for  it  was 
Archytas  who  worked  out  and  applied 
the  principles  of  the  pulley.  He  is  be- 
lieved to  have  been  the  first  student  to 
find  a  solution  of  the  problem  called 
the  "  duplication  of  the  cube,"  that  is, 
to  find  the  dimensions  of  a  cube  the 
volume  of  which  shall  be  twice  that  of 
a  given  cube. 

Plato  (429-348  B.C.)  was  a  contem- 
porary of  Archytas,  and  his  name  is 
associated  even  more  generally  with 
geometrical  science  than  that  of  his 
compatriot.  Plato  called  his  school  the 
"  Academy/'  and  the  underlying  prin-  piato 


INTRODUCTION  9 

ciple  of  his  theory  was  the  abstract  and  systematic  development  of 
geometric  science.  Plato  insisted  that  the  only  instruments  needed 
for  the  study  of  the  subject  were  the  straight  edge  and  the  com- 
passes, and  the  history  of  the  science  has  demonstrated  the  ac- 
curacy of  his  conclusions,  as  they  are  the  only  scientific  tools 
needed  for  all  elementary  work  in  the  science. 

Eudoxus  (408-355  B.C.)  studied  under  Plato  for  a  time,  and 
subsequently  did  some  independent 
work  in  the  science.  He  directed  his 
attention  chiefly  to  the  principles  of 
proportion  and  certain  methods  of 
proof,  to  which  we  shall  make  refer- 
ence later.  He  was  the  first  scientist 
to  begin  to  put  into  book  form  the 
mathematical  knowledge  of  his  time, 
and  may  properly  be  considered  the 
logical  forerunner  of  the  mathemati- 
cian Euclid.  Euclid,  who  was  a  teacher 
in  a  school  of  mathematics  in  Alex- 
andria, Egypt,  about  300  B.C.,  was  the  author  of  what  is  probably 
the  most  famous  book  on  geometry.  He  collected  and  arranged  all 
the  knowledge  of  the  science  down  to  his  own  time,  and  his  book 
still  stands  today  in  many  respects  as  a  final  authority  and  the 
background  of  the  entire  science.  Despite  the  fact  that  the  Egyp- 
tians and  Babylonians  first  developed  a  crude  knowledge  of  the 
subject,  it  is  perhaps  appropriate  that  the  Greeks,  whose  generations 
of  scientists,  culminating  in  Euclid,  gave  so  much  study  to  it,  should 
have  furnished  the  name  by  which  we  call  it.  The  Greek  word 
"ge"  meaning  the  earth,  and  "metron"  to  measure,  are  the  roots 
which  formed  our  name  for  the  science 

From  its  first  crude  beginning  in  the  necessity  for  measuring  the 
destruction  wrought  by  an  ancient  river,  its  instruments,  crude 
pegs  and  a  knotted  rope,  developed  and  applied  as  a  science  by  the 
mathematicians  of  five  centuries  before  the  Christian  era,  supple- 
mented and  enlarged  by  the  observations  and  discoveries  of  nearly 
twenty  centuries  of  research,  the  science  by  which  the  surveyors  of 
Egypt  located  their  boundaries  is  today  the  method  used  for  deter- 
mining the  power  of  a  battleship  or  the  contents  of  a  mountain  range. 


10  PLANE  GEOMETRY 

SUMMARY 
I  .  GEOMETRY  AMONG  THE  BABYLONIANS  AND  EGYPTIANS 

A.  DERIVATION  OF  THE  WORD  "GEOMETRY." 

B.  EVIDENCES  OF  KNOWLEDGE  OF  GEOMETRY. 

1.  Among  the  Babylonians. 

a.  Documentary  evidence. 
(I)  Clay  tablets. 
(II)  Talismans. 
(Ill)  Monuments. 

2.  Among  the  Egyptians. 

a.  Evidences  in  practical  life. 

(I)  Surveying. 

(II)  Reservoirs. 

(III)  Irrigation. 

(IV)  Pavements. 

6.  Evidences  in  religious  life. 

(I)  Orientation  of  temples. 
(II)  Pyramids, 
c.  Documentary  evidence. 
(I)  Ahmes  Papyrus. 
(II)  Hieroglyphics 
II.  GEOMETRY  AMONG  THE  GREEKS. 

A.  SOURCE. 

B.  SCHOOLS  THAT  CONTRIBUTED  TO  THE  DEVELOPMENT  OF  GEOMETRY. 

1.  Ionic  School,    a.  Thales  (640-546  B.C.). 

.  (a.  Pythagoras  (580-501  B.C.). 

2.  Pythagorean  School  ],     A  J  ,          IMOA  o«« 

(6.  Archytas  (430-365  B.C.). 

Q    Plot     •    Q  fc     i    fa-  Plato  (429-348  B.C.). 

3.  Platonic  School  i  »    «.  j          /^oorr        \ 

( 6.  Eudoxus  (408-355  B.C.) 

n   /-,        -i       f  o.  Hippocrates  (C.  440  B.C.). 

C.  Compilers  ]  ,    _  * 

(b.  Euclid  (C.  300  B.C.). 

BIBLIOGRAPHY 
ALLMAN,  G.  J.:  "Greek  Geometry  from  Thales  to  Euclid":  pp.  2,  3,  5,  7,  15, 

16,  22,  29,  41,  139-140,  143  (footnote)  and  154. 
BALL,  W.  W.  R.:  "A  Primer  of  the  History  of  Mathematics":  pp.  3-6,  8-14, 

32,  42-48. 
BALL,  W.  W.  R.:  "A  Short  History  of  Mathematics":  pp.  5-8, 15-16,  25-27, 

40,  54-63,  69-72. 

FINK,  K.:  "Brief  History  of  Mathematics":    pp.  190-214  (with  omissions). 
Gow,  J.:  "A  History  of  Greek  Mathematics":  pp.  176-178. 
BOYER,  J.  F.:  "HistoiredesMathematiques":  Chapters  I-V. 
CANTOR,  MORITZ:   "Vorlesungen  iiber  Geschichte  der  Mathematik":  Vol.  I, 

pp  17-52,  90-114,  134-146,  170-187,  201-277. 


INTRODUCTION  11 

EXERCISES.    SET  I.    BASED  UPON  HISTORIC  FACTS 

1.  From  the  derivation  of  the  word  geometry,  can  you  suggest 
any  studies  or  professions  in  which  geometry  may  be  applied? 

2.  What  gave  rise  in  the  first  place  to  the  art  and  eventually  to 
the  science  of  geometry? 

3.  From  the  little  told  you  in  the  foregoing  paragraphs  and  any 
references  you  may  have  read,  what  would  you  judge  to  be  the 
essential  difference  between  the  geometry  of  the  Egyptians  and 
the  geometry  of  the  Greeks? 

4.  Judging  from  the  character  of  the  Roman,  would  you  expect 
him  to  do  much  to  advance  the  science  of  geometry? 

5.  Among  the  formulas  given  in  Ahmes  Papyrus  for  determining 
areas  are  the  following:  I.  The  area  of  an  isosceles  triangle  equals 
half  the  product  of  the  base  and  one  of  the  equal  sides.    II.  The 
area  of  an  isosceles  trapezoid  equals  half  the  product  of  the  sum 
of  the  bases,  and  one  of  the  equal  sides. 


a.  Using  6,  61,  for  the  bases,  and  s  for  each  of  the  equal  sides 
write  an  algebraic  formula  for  each  of  these  areas,  A. 

b.  What  is  the  error  in  each  of  these  formulas? 

c.  Draw  figures  to  show  that  at  times  this  error  would  not 
matter  much. 

d.  Draw  figures  showing  cases  where  the  error  would  make  a 

considerable  difference.  ,/_  _ 

T     ,1  ... 

e.  In  the  accompanying  dia- 
gram find  j  ust  what  error  is  made     a 

(correct  to  tenths)  by  using  the  

Egyptian  formula.  k=9° 

6.  Another  formula  given  in  Ahmes  Papyrus  is  that  for  finding 
the  area  of  a  circle.     It  tells  you  to  multiply  the  square  of  the  radius 
by  !%.     What  value  must  the  Egyptian  then  have  assigned  to  ?r? 

7.  What  is  meant  by  saying  that  a  3-4-5  triangle  is  a  right 
triangle? 


12  PLANE  GEOMETRY 

8.  Show  by  knotting  a  piece  of  cord  so  that  the  parts  have  the 
ratio  3  to  4  to  5  how  the  Egyptian  "  rope-stretchers  "  obtained  their 
east-west  line.    (Stretch  the  cord  around  pins  on  a  board  and  after  it 
is  in  place  test  the  accuracy  of  the  method  with  your  right  triangle.) 

9.  Plato  and  his  school  interested  themselves  in  the  so-called 
Pythagorean  numbers.    Such  numbers  are  those  that  would  repre- 
sent the  lengths  of  the  sides  of  a  right  triangle. 
In  this  kind  of  triangle  they  must  be  such  that 

c2.     The  school  of  Plato  found  that 


+1,  ft,  and  (-J^)2  —  1  were  Pythagorean  numbers. 

a.  Verify  the  statement. 

b.  Find  ten  sets  of  Pythagorean  numbers. 

10.  Pythagoras  himself  found  that  n,  %(n2  —  1),  and  M(ft2+l) 
were  numbers  such  as  described  in  the  last  exercise.    Verify  this 
statement. 

11.  Bramagupta,  a  Hindu  writer  of  the  seventh  century,  gave 
P>  J£(gfq)>  and  ^2(3  ~q)  as  Pythagorean  numbers. 

a.  Give  various  values  to  p  and  q  to  test  his  statement. 

b.  Verify  his  statement. 

12.  In  the  Culvasutras,  a  Hindu  manuscript,  directions  for  con- 
structing a  right  angle  are  as  follows:    Divide  a  rope  by  a  knot 
into  parts  15  and  39  units  in  length  respectively,  and  fasten  the 
ends  to  a  piece  36  units  in  length. 

a.  Draw  a  diagram  to  show  what  is  meant  by  this. 

6.  Check  to  see  whether  these  are  Pythagorean  numbers. 

c.  Is  it  true  that  all  numbers  having  the  ratio  of  these  three  are 
Pythagorean  numbers? 

13.  Archimedes  proved  that  the  value  of  TT  lay  between  3)^  and 

How  does  this  compare  wiih  the  value  we  use  to-day? 


B.  A  FEW  ILLUSTRATIONS  OF  GEOMETRIC  FORM 

Before  we  begin  a  systematic  study  of  geometry,  let  us  see  if  we 
can  find  any  illustrations  of  the  kind  of  forms  about  which  we  hope 
to  learn  something.  Do  we  not  find  such  forms  in  nature?  We 


INTRODUCTION 


13 


recall  symmetrical  trees  and  conical  mountains;  we  think  of  a 
circular  moon,  spherical  raindrops  and  crystals  of  many  forms. 


Symmetrical  tree 


Conical  mountain 


Crystals 

EXERCISES.    SET  II.   ILLUSTRATIONS  OF  GEOMETRIC  FORMS 
14.  Make  a  list  of  some  geometric  forms  you  have  found  in 
nature,  under  the  following  heads:   spherical,  conical,  cylindrical, 
prismatic,  circular,  etc. 

Aside  from  natural  objects,  geometric  forms  continually  appear 
in  the  works  of  man.  The  building  and  room  in  which  we  are,  the 
furniture,  windows,  doorways,  all  are  geometric  in  form.  The 
familiar  objects  of  our  daily  life — coins,  boxes,  cylindrical  tubes, 
balls — all  illustrate  the  application  of  geometrical  principles. 


14  PLANE  GEOMETRY 

15.  As  in  the  preceding  exercise,  make  a  list  of  some  geometric 
forms  you  can  find  in  the  works  of  man. 

C.  MEANING  OF  GEOMETRIC  FORMS 

If  we  note  these  forms  carefully,  we  see  that  they  are  various 
combinations  of  the  simple  elements — points,  lines,  surfaces  and 
solids.  At  the  outset,  therefore,  we  should  be  sure  that  our  ideas 
about  these  elements  are  correct. 

First  let  us  consider  a  geometric  solid.  We  have  seen  cones  made 
of  wood,  and  others  of  ice  cream;  we  have  looked  into  a  well  and 
said  it  was  cylindrical;  we  have  watched  a  soap  bubble  and  called 
it  spherical.  So  we  see  that  it  is  the  shape  or  form,  and  not  the 
substance  of  which  an  object  is  made,  to  which  we  refer  in  speaking 
of  a  geometric  solid.  When  we  mention  a  sphere,  we  mean  the 
space  which  it  occupies  or  its  shape  without  reference  to  its  physical 
properties  or  the  material  of  which  it  is  made. 

If,  as  we  have  just  noted,  a  solid  is  a  limited  portion  of  space,  what 
limits  it?  How  is  a  solid  separated  from  the  rest  of  space?  The 
boundaries  of  a  solid  are  surfaces,  and  since  a  solid  is  identified, 
not  by  its  material,  but  only  by  its  shape,  so  must  its  boundary 
be  identified  by  its  shape.  A  chalk  box,  for  instance,  is  in  the 
form  of  a  prism,  i.e.,  the  space  it  occupies  is  a  geometric  prism. 
The  boundaries  of  this  prism  are  called  surfaces,  and  they  divide 
it  from  the  rest  of  space. 

The  surfaces  meet  and  form  lines.  The  edges  of  the  chalk  box 
are  referred  to  as  the  union  of  its  sides.  Now  if  we  think  of  the 
geometric  prism — the  space  occupied  by  the  box — the  intersection 
of  the  surfaces  are  lines. 

It  is  evident,  then,  that  lines  crossing  form  points.  A  limited 
portion  of  space  is  called  a  solid,  the  boundaries  of  a  solid  are  called 
surfaces,  the  intersections  of  surfaces  are  called  lines,  and  the  places 
where  lines  cross  are  called  points. 

We  see  thus  that  a  point  is  a  place  or  position,  and  can,  therefore, 
have  no  length,  breadth,  or  thickness.  For  convenience,  we  repre- 
sent a  point  by  a  dot  of  lead,  ink,  or  chalk.  Such  a  dot  is  obviously 
not  a  point,  because  it  has  some  size,  however  small  it  may  be,  but 
it  marks  a  location,  which  is  the  real  point. 


INTRODUCTION  15 

If  we  could  imagine  a  point  to  move,  we  would  call  its  path  a 
line.  A  line,  then,  would  have  no  width,  but  it  would  have  length. 
If  we  could  now  imagine  a  line  to  move,  not  along  itself,  we  would 
say  it  generated  a  surface.  Again,  the  surface  would  have  only 
length  and  width,  but,  of  course,  no  depth.  If  we  consider  a  surface 
to  move,  not  along  itself,  it  would  form  a  solid,  which  would  have 
three  dimensions. 

In  our  study  of  geometry  we  shall  have  to  deal  with  straight  and 
curved  lines.  Let  us  note,  then,  that  a  straight  line  is  one  which 
is  fixed  (or  determined)  by  any  two  of  its  points,  and  a  curved  line 
is  one  no  part  of  which  is  straight.  Throughout  the  book,  it  is  to 
be  understood  that  when  the  word  line  is  used  without  a  qualifying 
adjective,  a  straight  line  is  designated.  A  line  is  of  indefinite 
length,  so  that  when  we  wish  to  refer  to  a  limited  portion  of  a  line 
we  shall  call  it  a  sect.  All  the  facts  with  which  we  shall  be  concerned 
for  a  time  will  be  those  relating  to  a  single  plane.  A  plane  is  a 
surface  such  that  if  any  two  points  in  it  be  connected  by  a  straight 
line,  that  line  lies  wholly  within  the  surface. 

EXERCISES.    SET  III.    MEANING  OF  GEOMETRIC  FORMS 

16.  If  a  series  of  600  points  were  put  within  an  inch  would  they 
form  a  line? 

17.  A  machine  has  been  manufactured  which  will  rule  10,000  dis- 
tinct lines  within  the  space  of  one  inch.     Are  these  lines  geometric? 

18.  Fold  over  a  piece  of  paper.  What  will  the  crease  represent? 

19.  If  oil  is  poured  on  water,  of  what  material  is  the  surface 
formed? 

20.  Put  your  foot  in  a  heap  of  snow  and  quickly  withdraw  it. 
Is  the  impression  that  is  left  a  physical  or  geometric  solid? 

21.  If  I  place  a  piece  of  red  paper  on  a  blue  one,  what  is  the 
color  of  the  surface  between  them? 

22.  If  1000  geometric  surfaces  were  placed  one  on  top  of  the 
other  would  a  geometric  solid  be  formed? 

23.  Is  a  cake  of  ice  a  geometric  solid? 

24.  (a)  Make  a  list  of  some  things  in  life  which  are  referred  to 
as  points.    (6)  How  many  of  these  are  geometric  points? 

25.  (a)  Make  a  list  of  some  things  in  life  which  are  referred  to 
as  lines.     (6)  How  many  of  these  are  geometric  lines? 


16 


PLANE  GEOMETRY 


D.  SUGGESTIONS  OF  A  FEW  USES  OF  GEOMETRY 

We  have  reviewed  briefly  the  historical  development  of  geometry 
and  have  called  to  mind  illustrations  of  geometric  forms,  both  in 
nature  and  in  manufactured  articles  and  have  clarified  our  ideas 
of  these  forms. 

Let  us  now  consider  a  few  of  the  uses  of  geometry.  The  subject 
grew  out  of  the  need  of  land-measuring.  Hence,  historically  at 
least,  surveying  is  the  first  known  use  of  geometry.  The  following 

illustrations  show  some 
of  the  things  that  we 
ought  soon  to  be  able  to 
do.  Laying  out  bounda- 
ries of  property  so  that 
the  owner  shall  have  his 
just  share,  or  finding  the 
areas  of  pieces  of  ground, 

Diagram  i  are    problems   requiring 

practical    mensuration 


A  B 

Diagram  II 


Diagram  III 

Transit 

about  which  we  shall  soon  study.     Finding  the  distances  between 
inaccessible  points,  as  from  X  to  Y  in  diagram  I,  across  rivers  and 


INTRODUCTION  17 

over  swamps,  as  in  Exercises  111-113,  and  heights  of  objects  as  in 
diagrams  II  and  III,  are  all  geometric  problems  such  as  are  included 
in  surveying.  We  have  all  seen  men  in  our  streets  with  transits 
on  tripods.  (A  transit  is  an  instrument  for  measuring  angles.) 
Geometry  is  necessary  to  solve  the  problems  for  which  they  are 
getting  the  data. 

EXERCISES.    SET  IV.    MENSURATION 

26.  Make  a  list  of  mensuration  formulas  with  which  you  are 
already  familiar. 

27.  Find  the  area  of  the  following  piece  of  property  (three  lots). 
The  measurements  taken  by  a  surveyor  are  noted  on  the  diagram. 

Other  surveying  problems 


will  be  found  later  in  the  book. 


65' 


We  do  not  yet  know  enough 
geometry  to  solve  many  such 
problems.  50'  95' 

Another  use  of  geometry  that  quickly  comes  to  the  mind  is 
designing.  We  mark  the  use  of  geometric  design  in  parquet  floors, 
linoleums,  tilings,  wall  and  ceiling  papers,  grill-work,  stained-glass 
windows,  arches,  and  in  similar  objects.  By  learning  how  to 
make  five  fundamental  constructions  (Exercises  28-32,  inc.), 
we  shall  be  able  to  combine  them  into  many  geometric  designs,  and 
thus  get  a  clearer  idea  of  one  of  the  uses  of  geometry.  The  reasons 
why  these  constructions  are  correct,  and  more  elaborate  work 
in  design,  must  be  postponed  until  later  in  the  text. 

EXERCISES.    SET  V.    CONSTRUCTIONS— DESIGNING 

All  the  constructions  in  these  exercises  are  to  be  made  with  the 
use  of  compasses  and  unmarked  straight  edge  only.  The  pupil  is 
reminded  that  unfamiliar  technical  terms  will  be  found  by  refer- 
ring to  the  index. 

In  general,  in  geometry,  auxiliary  lines  (those  needed  only  as 
aids)  are  indicated  by  dotted  lines,  preferably  light. 

28.  From  a  given  point  on  a  given  straight  line  required  to  draw 
a  perpendicular  to  the  line. 

Let  AB  be  the  given  line  and  P  be  the  given  point. 

It  is  required  to  draw  from  P  a  line  perpendicular  to  AB. 
2 


18 


PLANE  GEOMETRY 


\X 


Y  / 


With  P  as  center  and  any  convenient  radius  strike  arcs  cutting 

AB  at  X  and  F. 

With  X  as  center  and  XFas  radius 
strike  an  arc,  and  with  F  as  center 
and  the  same  radius  strike  another 
arc,  and  call  one  intersection  of  the 
arcs  C. 

With  a  straight  edge  draw  a  line 

J— g      through  P  and  C,  and  this  will  be  the 

perpendicular  required. 
29.  From  a  given  point  outside  a  given  straight  line  required 
to  let  fall  a  perpendicular  to  the  line. 

Let  AB  be  the  given  straight  line  and  P  be  the  given  point. 
It  is  required  to  draw  from  P  a  line  perpendicular  to  AB. 
With  P  as  center  and  any  convenient 
radius  describe  an  arc  cutting  AB  at  X 
andF. 

With  X  as  center  and  any  convenient 
radius  describe  an  arc,  and  with  F  as 
center  and  the  same  radius  describe  an- 
other arc,  and  call  one  intersection  of  the 
arcs,  C. 

With  a  straight  edge  draw  a  straight 
line  through  P  and  C,  and  this  will  be 

the  perpendicular  required. 

It  is  interesting  to  test  the  results  in  Ex- 
ercises 28  and  29  by  cutting  the  paper  and 
fitting  the  angles  together. 

30.  Required  to  bisect  a  given  sect. 
Let  AB  be  the  given  sect. 
—       It  is  required  to  bisect  AB. 

With  A  as  center  and  AB  as  radius  de- 
scribe an  arc,  and  with  B  as  center  and  the 
same  radius  describe  another  arc. 

Call  the  two  intersections  of  the  arcs  X 
andF. 

Draw  the  straight  line  XY. 
Then  XY  bisects  the  sect  AB  at  the  point  of  intersection  M. 


INTRODUCTION 


19 


31.  From  a  given  point  on  a  given  line  required  to  draw  a  line 
making  an  angle  equal  to  a  given  angle. 
Let  P  be  the  given  point  on  the  given 
linePQ,  and  let  angle  A  OB  be  the  given 
angle. 

What  is  now  required? 
With  0  as  center  and  any  radius  de- 
scribe an  arc  cutting  A  0  at  C  and  BO  at  D. 
With  P  as  center  and  OC  as  radius  de- 
scribe  an  arc  cutting  PQ  at  M. 

With  M  as  center  and  CD  as  radius  de- 
scribe an  arc  cutting  the  arc  just  drawn  at 
N,  and  draw  PN. 

Then  angle  MPN  is  the  required  angle. 

32.  Required   to  bisect  a  given 
angle. 

Let  A  OB  be  the  given  angle. 
It  is  required  to  bisect  the  angle 
AOB. 

With  0  as  center  and  any  conve- 
nient radius  strike  an  arc  cutting 
OA  at  X  and  OB  at  Y. 
With  X  as  center  and  the  sect  XY  as  radius  strike  an  arc,  and 
with  Y  as  center  and  the  same  radius  strike  an  arc,  and  call  one 
point  of  intersection  of  the  arcs  P. 
Draw  the  straight  line  OP. 
Then  OP  is  the  required  bisector. 
33.  Make  constructions  similar  to  the  following : 

Suggestion:  AB=OB. 


20 


PLANE  GEOMETRY 


34.  Draw  the  fol- 
lowing figure. 


35.  Make  a   con- 
struction similar  to  a. 


d36.*  Copy  b. 


a  b 

37.  From  a  study  of  Exercise  28,  suggest  how  to  erect  a  per- 
pendicular at  the  end  of  a  sect. 

38.  On  a  given  sect  construct  a  square. 

39.  These  figures  show  a  parquet  floor  design,  and  one  of  the 
units  of  the  design  enlarged.    Construct  figures  similar  to  these. 

ABCD  is  a  square, 
and  X,  Y,Z,  and  IF  are 
the  mid-points  of  the 
semidiameters  OE,  OF, 
OG,  OH,  respectively. 


G 


H 


A  E 

40.  Make  a  construction  similar  to  the  ad- 
joining figure. 

The  vertices  of  the  square  are  used  as  centers 
for  four  of  the  arcs. 

The  radius  equals  one  side  of  the  square. 


41.  Make  a 
construction 
similar  to  a. 


/ 


42.  Make  a 
construction 
similar  to  b. 


a 


*  As  here,  d  will  be  prefixed  to  any  exercise  which  the  student  is  likely  to 
find  difficult  at  this  stage. 


INTRODUCTION 


21 


d43.   Make   a  construction  similar  to  the  following: 

The  figure  is  based  on  an  equilateral  triangle,  the  centers  of  the 
interior    arcs  being    the  midpoints    of    radii 
drawn  to  the  vertices  of  the  equilateral   tri- 
angle inscribed  in  a  circle  (i.e.,  having  its  ver- 
tices on  the  circle) . 

Note:  See  Fig.  2,  exercise  33. 

N.  B. — The  design  shown  in  MableSykes,  "Source- 
Hook  of  Problems  for  Geometry,"  page  160,  II,  5  (Fig. 
138a),  shows  a  good  application  of  exercise  43. 

44.  Make  a  construction 
similar  to  a. 

d45.  Make  an  ornamental 
design  similar  to  b.  The  circle 
is  divided  into  how  many 
equal  arcs?  How  many  de- 
grees in  each  central  angle? 

What  kind  of  triangle  is  formed  by 
two  consecutive  radii  and  the  sect 
j  oining  their  ends?  What  other  method 
does  this  suggest  of  dividing  a  circle 
into  six  equal  arcs? 

46.  Make  an  ornamental  drawing 
similar  to  the  one  in  the  accompanying 
figure.  Describe  the  construction. 

Suggestion:  First  draw  an  equilateral  poly- 
gon with  six  sides  in  a  circle. 

47.  Construct  a  six- 
__j_  pointed  star. 

48.  Bisect   each   of 
the   four  right  angles 
formed   by   two  lines 
intersecting  each  other 
at  right  angles. 

49.  Make  construc- 
tions   similar    to    the 
following : 


22  PLANE  GEOMETRY 

50.  Make  constructions  similar  to  the  following: 


In  such  figures  artistic  patterns  may  be  made  by  coloring  vari- 
ous portions  of  the  drawings. 

In  this  way  designs  are  made  for  stained-glass 
windows,  oil-cloth,  colored  tiles,  and  other  dec- 
orations. 

51.  Draw  a  sect  of  any  convenient  length, 
and  upon  it  construct  a  design  similar  to  the 
one  in  the  figure. 

52.  On  a  line  LM  take  a  sect  AB.    Divide  it 
into  8  equal  parts.  With  your  compasses  make 

an  ornamental  scroll  as  shown 
in  the  diagram. 

53.  Using  the  hints  given, 
make  a  copy  of  the  accompany-  L 
ing  outline  drawing  of  a  Gothic 
window.   The  arc  BC  is  drawn 

with  A  as  center  and  AB  as  radius.  The 
small  arches  are  described  with  A,  D, 
and  B  as  centers  and  AD  as  radius. 
The  center  P  is  found  by  taking  A  and 
B  as  centers  and  AE  as  radius.  How 
may  the  points  D,  E,  and  F  be  found? 

54.  In  many  different  machines,  such 
as  the  sewing  machine,  printing  press, 
A  F  D  E  B  etc.,  there  is  a  wheel  called  a  cam,  which 
is  used  to  modify  the  motion  of  the  machinery.  Cams  are  con- 
structed in  various  shapes  and  dimensions,  depending  upon  the  use 
for  which  they  are  designed.  The  figure  shows  the  method  of  draw- 
ing the  pattern  of  a  heart-shaped  or  "  uniform-motion  "  cam.  Let 


INTRODUCTION 


23 


Divide  A  B  into  eight  equal 
.  .  .  ,  B,  draw  circles  with 


the  "  throw  "  be  AB  and  the  center  0. 
parts  at  C,  D,  etc.    Through  A,  C,  D, 
centers  at  0.    Draw  sects  dividing  the 
angular  magnitude  around  0  into  six- 
teen  equal    parts.     Beginning  at  A, 
mark  the  points  where  the  consecutive 
circles  and  consecutive  sects  intersect, 
and  through  these  points  draw  a  smooth 
curve,  as  in  the  figure. 

Draw  such  a  cam  with  AB  equal  to  a 
given  sect  m,  and  OA  equal  to  a  given 
sect  n. 

(Taken  with  modifications  from 
Stone-Millis,  Elementary  Plane  Geometry.) 

55.  Select  and  copy  some  geometric  design. 

56.  Make  an  original  design  based  on  the  fundamental  construc- 
tions given  in  exercises  28-32. 

Pupils  particularly  interested  in  this  part  of  the  work  are 
referred  to:  Sykes,  Mabel,  "Source  Book  of  Problems  in  Geo- 
metry." (Pub.  Allyn  and  Bacon.) 

Geometry  is  used  in  architecture.  Whether  the  architect  is 
drawing  the  plans  for  an  ordinary  dwelling-house  or  a  massive 
cathedral,  he  is  constantly  concerned  with  geometric  forms  and 
constructions.  Consider  for  a  moment  what  problems  of  this 
character  must  have  confronted  the  architect  of  some  large  building 
in  our  community. 

The  list  of  the  direct  uses  of  geometry  would  be  very  long  if 
complete.  In  a  few  sentences  let  us,t  therefore,  simply  enumerate 
a  few  more  miscellaneous  suggestions  for  its  uses.  Problems 
scattered  throughout  this  book  show  more  concretely  how  geometry 
is  used  in  the  cases  enumerated.  In  making  all  kinds  of  diagrams, 
reducing  and  enlarging  maps,  the  principles  of  geometry  are  applied. 
In  engineering,  geometry  is  needed  for  such  matters  as  laying  out 
railroads,  and  planning  the  constructions  of  machines,  bridges, 
and  tunnels;  and  in  astronomy,  ascertaining  the  altitude  of  stars 
and  similar  problems  require  geometric  principles. 


24  PLANE  GEOMETRY 

EXERCISES.    SET  VI.    SOME  OTHER  USES  OF  GEOMETRY 

57.  State  any  other  uses  of  geometry  which  you  know. 

58.  At  the  entrance  to  New  York  Harbor  is  a  gun  having  a 
range  of  12  mi.    Draw  a  line  inclosing  the  range  of  fire,  using  any 
convenient  scale. 

59.  Two  forts  are  placed  on  opposite  sides  of  a  harbor  entrance, 
13  mi.  apart.    Each  has  a  gun  having  a  range  of  10  mi.    Draw  a 
plan  showing  the  area  exposed  to  the  fire  of  both  guns,  using  any 
convenient  scale. 

60.  Make  an  accurate  diagram  of  a  tennis  court  or  a  foot-ball 
field  noting  all  lime  lines. 

d61.  Draw  to  a  convenient  scale  a  plan  of  the  ground  floor  of 
your  school  building. 

E.  THE  BASIC  PRINCIPLES  OF  GEOMETRY 

As  in  all  scientific  work  of  an  exact  nature,  the  discoveries  in 
geometry  rest  upon  a  few  basic  principles.  These  may  be  classified 
under  three  heads :  definitions,  axioms,  and  postulates. 

You  all  can  probably  recall  having  heard  people  argue  most 
heatedly  about  some  question  and  reach  no  conclusion  at  all.  This 
is  often  the  case  simply  because  when  two  people  argue,  they  very 
often  do  so  without  having  clearly  in  mind  the  conditions  about 
which  they  are  arguing.  In  all  debates  or  discussions  it  is  essential 
that  we  start  with  the  same  premises,  and  our  work  in  geometry 
should  help  us  to  learn  to  collect  our  premises  in  orderly  fashion. 

The  premises  upon  which  the  early  parts  of  geometry  rest  are 
to  a  great  extent  definitions,  and  it  is  therefore  very  necessary  that 
we  have  a  clear  image  and  definition  of  each  new  technical  term 
we  meet.  The  wording  of  our  definitions  may  differ,  but  the  con- 
tent must  be  the  same.  Every  good  definition  should  include  all 
that  may  fall  under  a  particular  class,  and  clearly  exclude  all  that 
does  not  fall  under  that  class.  Suppose,  for  instance,  we  want  to 
define  the  word  botany.  We  might  say,  to  begin  with,  that  it  is 
a  science — but  we  have  not  differentiated  it  from  the  physical 
sciences,  so  we  say  it  is  a  natural  science.  But,  then,  so  is  zoology. 
Hence  it  is  necessary  to  differentiate  still  further,  and  say  it  is  the 
natural  science  which  deals  with  plant  life.  Now  have  we  fully 


INTRODUCTION  25 

and  finally  defined  it?  Can  you  possibly  think  of  any  science — 
now  that  it  is  thus  defined — with  which  it  can  be  confused?  Make 
all  possible  tests,  and  if  you  find  that  it  cannot  be  confused  with 
any  other  science,  well  and  good, — then  we  have  found  an  accept- 
able definition. 

Throughout  this  book  no  word  will  be  defined  until  we  are  ready 
to  make  use  of  it,  but  then  it  will  be  our  duty  to  see  the  word  in 
its  full  meaning. 

From  the  knowledge  we  have  of  algebra,  we  already  know  what 
some  of  the  axioms  are,  and  some  uses  to  which  they  may  be  put, 
though  we  may  not  have  defined  the  word  "  axiom."  Some  of  the 
axioms  for  which  we  shall  have  immediate  use  are : 

1.  The  sums  of  equals  added  to  equals  are  equal. 

Example  If  5 =5 

and  a=b 
then  5+0^5+6. 

2.  The  remainders  of  equals  subtracted  from  equals  are  equal. 

Example  If  a=c 

and  b=d 
then  a—b  =  c—d 

3.  The  products  of  equals  multiplied  by  equals  are  equal. 

Example  If  a=x 

and  b=y 
then  ab=xy. 

4.  The  quotients  of  equals  divided  by  equals  are  equal. 

Example  If  x=y 

and  m=p 

then  -^ 

m     p 

Cases  in  which  the  divisor  is  zero  will  not  be  considered  in  this 
text. 

5.  A  quantity  may  be  substituted  for  its  equal  in  a  statement  of 
equality  or  inequality. 

Example  If  x  =  5  If  a = b 

and  x+y=7  and  2a+5>a+2 

then  by  substitution  5+y=7  then  26 + 5  >  a + 2 


26  PLANE  GEOMETRY 

6.  Two  quantities  which  are  equal  to  equal  quantities,  are  equal  to 
each  other. 

Example  If  a=b 

b=c 

and  c=d, 
then  a=d 

7.  The  whole  is  equal  to  the  sum  of  Us  parts. 

T.  a  .  a  .  a 

Example  §    3    6^ 

NOTE:  "part"  is  here  used  in  the  sense  of  a  common  or  vulgar 
fraction. 

Other  axioms  will  be  stated  as  we  need  them.  Thus  we  see  that 
an  axiom  is  the  statement  of  a  general  mathematical  truth  which  is 
granted  without  any  proof. 

A  postulate  is  simply  a  geometric  axiom.  That  is,  it  is  the  state- 
ment of  a  geometric  truth  which  is  granted  without  any  proof. 

We  shall  now  attempt  to  formulate  a  few  such  truths. 

EXERCISES.    SET  VII.    ILLUSTRATIONS  OF  POSTULATES 

62.  Why  is  it  shorter  to  cut  across  a  field  than  to  go  around  it? 

g63.*  How  many  pairs  of  roots  are  there  when  two  simultaneous 
linear  equations  are  solved?  What  is  the  graphic  explanation  of 
this? 

g64.  Why  is  it  that  in  making  the  graph  of  a  linear  equation, 
such  as  x-\-y  =  13,  we  need  to  plot  but  two  points,  and  that  a  third 
point  may  be  used  to  check  the  correctness  of  our  work? 

65.  Why  is  it  that  the  Panama  Canal  is  a  great  advantage  over 
the  route  formerly  used  to  reach  a  point  on  the  western  coast  of 
South  America  from  the  West  Indies? 

66.  Why  is  it  that  in  putting  up  a  croquet  set  all  one  needs  to 
do  to  get  the  wickets  in  line  with  the  stakes  is  to  tie  a  string  tightly 
to  one  stake  and  stretch  it  and  fasten  it  to  the  other  stake? 

*  As  here,  g  will  be  prefixed  to  any  exercise  in  this  text  which  presupposes 
an  acquaintance  with  the  graph.  The  pupil  is  here  referred  to  M.  Auerbach, 
An  Elementary  Course  in  Graphic  Mathematics  (Allyn  and  Bacon),  pp.  29-31, 
for  review,  and  previous  pages  in  the  same  if  the  subject  is  new  to  him. 


INTRODUCTION  27 

67.  What  does  the  bricklayer  do  to  get  a  row  of  bricks  in  a 
straight  line?    Why?    Does  the  gardener  do  anything  similar  to 
this? 

68.  Point  out  which  of  the  following  postulates  upholds  each 
of  your  answers  to  exercises  62-67. 

POSTULATES  OF  THE  STRAIGHT  LINE  NEEDED  IN  PROOFS 

1.  Two  intersecting  straight  lines  determine  a  point. 

2.  Two  points  determine  a  straight  line. 

3.  A  straight  line  is  the  shortest  distance  between  two  points. 

69.  How  often  can  two  straight  lines  intersect? 

70.  Use  your  answer  to  the  last  question  to  state  postulate  1  in 
another  way. 

71.  How  many  straight  lines  can  be  drawn  between  two  points? 

72.  Use  your  answer  to  the  last  question  to  state  postulate  2  in 
another  way. 

73.  Give  at  least  one  good  illustration  of  how  each  of  the  three 
postulates  mentioned  may  be  used  in  practical  life. 

We  have  used  several  other  postulates  in  making  some  of  the 
constructions  on  pages  13  to  18.  They  are:  (1)  A  sect  may  be 
produced  indefinitely.  (2)  A  circle  may  be  described  with  any  point 
as  center  and  any  sect  as  radius.  (3)  A  point  and  direction  determine 
a  straight  line.  But  these  are  so  exceedingly  obvious  that  we  shall 
not  feel  obliged  to  quote  them. 

DEFINITIONS 

An  angle  is  the  opening  between  two  lines.  The  lines  are  called 
the  sides  of  the  angle,  and  the  point  at  which  they  meet  the  vertex. 


An  angle  may  be  named  in  any  one  of  three  ways  as  <£  A  in 
Fig.  1  where  there  is  no  danger  of  confusion,  or  as  in  Fig.  2  *£ABC, 
<£ABX,  ^XBC  where  there  are  several  angles  (the  vertex  always 
being  read  second),  or  again  as  in  Fig.  3  <£ca, 


28  PLANE  GEOMETRY 

1.  Kinds  of  angles  defined  according  to  individual  size. 

A  straight  angle  is  one  whose  sides  run  in  opposite  directions  so 

^T\ as  to  form  a  straight  line. 

^~~  °  ^AOB  is  a  straight  angle. 

A  right  angle  is  one-half  a  straight 
angle. 

<£AOB   and    <£BOC    are    right 
angles. 

An  acute  angle  is  less  than  a  right 
angle. 


R  •£ tfST  is  an  acute  angle. 

An  obtuse  angle  is  greater  than  a  right  angle  but  kss  than  a  straight 
angle. 

is  an  obtuse  angle.  \ 


O  x 

2.  Kinds  of  angles  defined  according  to  sums. 
Complementary  angles  are  two  whose  sum  is  a  right  angle. 
Supplementary  angles  are  two  whose  sum  is  a  straight  angle. 

EXERCISES.    SET  VIII.    SUMS  OF  ANGLES  IN  PAIRS 
74.  Construct:   (a)  two  complementary  angles  whose  ratio  is  1 
to  3;  3  to  5. 

(6)  Two  supplementary  angles  whose  ratio  is  1  to  3;  3  to  5. 

MEASUREMENT  OF  ANGLES 

There  are  three  systems  of  measurement  of  angles.  The  one 
probably  known  to  most  of  us  is  the  sexagesimal  system,  and  was 
mentioned  on  page  6.  It  divides  the  entire  angular  magnitude 
about  a  point  into  360  parts,  each  of  which  is  called  a  degree;  these 
again  are  divided  into  sixtieths,  each  of  which  is  called  a  minute, 
and  each  minute  is  again  divided  into  sixtieths,  each  of  which  is 
called  a  second. 

The  size  of  an  angle  then  depends  upon  the  amount  of  opening 
between  its  sides.  The  amount  of  opening  depends  upon  the 


INTRODUCTION 


29 


amount  that  one  side  has  to  revolve  to  bring  it  into  the  position 
of  the  other,  and  the  greater  that  amount  the  greater  the  angle. 
Thus    in   these    com- 
passes the  first  angle 
is    smaller    than    the 
second,  which  is  also 
smaller  than  the  third. 
The  length  of  the  sides 
has  nothing  to  do  with 
the  size  of  the  angle. 

A  special  instrument,  called  a  protractor,  is  frequently  used 
for  angle  measurements.  The  figure  given  below  represents 
one  form  of  the  protractor.  By  joining  the  notch  O  of  the 
protractor  to  each  graduation  mark  we  obtain  a  set  of  angles 
at  0,  each  usually  representing  an  angle  of  one  degree. 

To  measure  a  given  angle  with  the  protractor,  place  the  notch 
of  the  protractor  at  the  vertex  of  the  angle,  and  the  base  line  along 

one  side  of  the  angle.  The 
other  side  of  the  angle  then 
indicates  on  the  protractor 
the  number  of  degrees  in 
the  angle.  Thus  the  angle 
AOB  contains  50°. 

To  draw  an  angle  of  a 
given  number  of  degrees, 
A    place  the  base  of  the  pro- 
tractor along  a  straight  line  and  mark  on  the  line  the  position  of  the 
notch  0.    Then  place  the  pencil  at  the  required  graduation  mark, 
and  (after  removing  the  protractor)  join  the  point  so  marked  to  0. 

EXERCISES.    SET  IX.    MEASUREMENT  OF  ANGLES 

75.  Show  how  a  fan  can  be  used  to  illustrate  the  idea  of  angular 
magnitude. 

76.  What  kinds  of  angles  are  formed  by  the  hands  of  a  clock  at 
(1)  two  o'clock,  (2)  three  o'clock,  (3)  four  o'clock,  (4)  six  o'clock? 

77.  What  kind  of  angle  is  equal  to  (1)  its  complement,  (2)  its 
supplement? 


30  PLANE  GEOMETRY 

78.  What  kind  of  angle  is  less  than  its  supplement? 

79.  What  angle  is  7°  less  than  one-third  its  complement? 

d80.  State  as  a  formula  (1)  the  number  of  degrees  in  the  com- 
plement of  the  supplement  of  any  angle  a,  (2)  the  number  of 
degrees  in  the  supplement  of  the  complement  of  angle  a. 

81.  Two  supplementary  angles  are  in  the  ratio  of  7  to  2.    Find 
the  number  of  degrees  in  each. 

82.  If  four  lines  a,  6,  c,  d,  are  drawn  from  a  point  0  in  the  order 
given,  so  that  a  is  perpendicular  to  c,  and  b  is  perpendicular  to  d, 
find  ^ad  if  ^bc  is  60°.    (See  definition,  page  31.) 

83.  Three  angles  together  make  up  the  angular  magnitude  about 
a  point.    The  first  is  10°  greater  than  the  second,  and  the  second 
is  17°  greater  than  one-half  the  third.    How  many  degrees  in  each? 

84.  Find  that  angle  whose  supplement  is  eight  times  its  comple- 
ment.   Is  it  possible  to  find  one  whose  supplement  is  one-eighth 
of  its  complement? 

85.  How  many  degrees  in  the  angle  which  exceeds  one-third  its 
complement  by  15°? 

86.  Find  the  number  of  degrees  in  the  angle  whose  excess  over 
its  complement  is  one-fourth  the  difference  between  its  complement 
and  itself. 

87.  Construct  two  straight  angles.    Cut  one  out  and  place  it 
on  the  other.    What  can  you  say  of  them?    Do  you  think  it  is 
true  of  all  straight  angles? 

88.  Construct  two  right  angles,     (a)  How  is  each  related  to  a 
straight  angle?    (b)  How  are  they  related  to  each  other?    Why? 

89.  Using  a  protractor  (a)  construct  three  angles  each  of  which 
is  the  complement  of  20°.    (b)  Construct  two  angles  each  of  30°. 
Construct   their  complements,      (c)  What   conclusions   can  you 
draw  from  (d)  and  (6)?    (d)  Why? 

90.  Do  the  same  as  you  did  in  exercise  89  for  the  supplements 
of  50°  and  60°. 

Corollary.  A  truth  that  is  directly  derived  from  another  is  called 
a  corollary.  The  conclusions  drawn  in  exercises  88-90  are  called 
corollaries,  since  they  are  all  directly  derived  from  the  conclusion 
drawn  in  exercise  87. 


INTRODUCTION 


31 


POSTULATES  OF  THE  ANGLES 

1.  All  straight  angles  are  equal. 

COT.  1.    All  right  angles  are  equal.    Why? 

Cor.  2.    Complement?  of  the  same  angle  or  equal  angles  are 

equal.    Why? 
Cor.  3.    Supplements  of  the  same  angle  or  equal  angles  are 

equal.    Why? 

Perpendicular.    A  ptrpendicuLar  is  a  line  that  meets  another  at 
right  angles. 

3.  Kinds  of  angles  defined  according  to  relative  position. 

Adjacent  angles  are  two  that  have  a 
common  vertex  and  a  common  side  lying 
between  them. 

and  <£BOC  are  adjacent. 


EXERCISES.    SET  X.    RELATIVE  POSITION  OF  ANGLES 

91.  Why  is  it  that  <£AOB  and  <£AOC  are  not  adjacent  angles? 

92.  Can  you  name  other  angles  in  the  diagram  which  are  not 
adjacent? 

93.  Tycho  Brahe  (1546-1601),  a  Danish  nobleman  who  built 
and  operated  the  first  astronomical  observatory,  in  his  earliest 

observations  used  a  quadrant 
for  measuring  the  altitudes 
of  stars,  or  their  angular  dis- 
tances above  the  horizon. 
Show  that  when  the  instru- 
ment was  held  in  a  vertical 
plane,  and  the  sights  A  and 
B  aligned  with  the  star  S,  the 
altitude  of  the  star  was  de- 
termined by  observing  the 
angle  CAD. 

(Taken  from  Stone  Millis, 
Elementary  Plane  Geometry.) 

94.  Make  such  a  quadrant 
of  cardboard  or  wood  and 


32 


PLANE  GEOMETRY 


use  the  method  of  exercise  93  to  find  the  elevations  of  objects  in 
the  neighborhood  such  as  trees,  hills,  steeples,  telephone-poles,  etc. 

Vertical  angles  are  those  which 
are  so  placed  that  the  sides  of  each 
are  the  prolongations  of  the  sides 
of  the  other.  OX  is  the  prolonga- 
tion of  BO,  and  OY  is  the  prolon- 
gation of AO.  Therefore  <£  YOX 
and  <£AOB  are  vertical  angles. 


EXERCISES.    SET  X  (continued) 

95.  Name  two  other  angles  in  the  preceding  figure  which  are 
vertical,  and  tell  why  they  are  vertical. 

96.  Classify  angles  according  to  (1)  individual  size,  (2)  relative 
size,  (3)  relative  position. 

97.  (a)  Draw  two  complemen- 
tary-adjacent angles. 

(6)  Draw  two  angles  of  the  same 
size  as  those  in  (a)  but  not  adjacent. 

(c)  Draw  two  supplementary- 
adjacent  angles. 

(d)  Draw  two  non-adjacent  sup- 
plementary angles. 

98.  In  the  accompanying  diag- 
ram select  those  angles  which  are 
straight,  right,  acute,  obtuse,  com- 
plementary, supplementary,  adja- 
cent, and  vertical. 


99.    Read  the    angle   which  is  equal 


to: 


O 


(a)   $.AOB+$BOC. 

(6) 
(c) 
(d) 


INTRODUCTION 


33 


100.  Construct  an  angle  equal  to: 


(a)  The  sum  of  these  three  angles. 

(6)  The  sum  of  $.PQR  and  $XYZ  less 


101.  Show  that  sects  bisecting  two  complementary-adjacent 
angles  form  an  angle  of  45°. 

102.  What  kind  of  angle  do  sects  bisecting  two  supplementary- 
adjacent  angles  form?    Prove  your  answer. 

103.  The  following  will  illustrate  the  unreliability  of  observation 
and  the  need  of  logical  proof. 

(Subdivisions  (a)  through  (/)  were  taken  fromWentworth-Smith, 
Plane  Geometry,  and  (g)  through  (i)  from  Hart  and  Feldman, 
jfr  Xj  Plane  Geometry.) 

(a)  Estimate  which  is  the  longer  sect, 
AB  orXY  ,  and  how  much  longer.  Then 
test  your  estimate  by  measuring  with  the     A  C  B 

compasses  or  with  a  piece  of  paper  carefully 
marked. 

(6)  Estimate  which  is  the  longer  sect,  ~AB 
or  CZ>,  and  how  much  longer.  Then  test  your 
estimate  by  measuring  as  in  (a). 

(c)  Look  at  this  figure  and  state  whether 


. 


D 


A  B  and  CD  are  both  straight 
ines.  If  one  is  not  straight, 
which  one  is  it?  Test  your  an- 
swer by  using  a  ruler  or  the 
folded  edge  of  a  piece  of  paper. 


(d)    Look   at  this   figure   and   state     A  ////////////////////, 


whether  AB  and  CD  are  the  same  dis- 
tance apart  at  A  and  C  as  at  B  and  D. 
Then  test  your  answer  as  in  (a)  . 
3 


34 


PLANE  GEOMETRY 


(e)  Look  at  this  figure  and  state  whether  AB  will,  if  prolonged, 
lie  on  CD.  Also  state  whether  WX  will,  if  prolonged,  lie  on  YZ. 
Then  test  your  answer  by  laying  a  ruler  along  the  lines. 


A  (/)  Look  at  this  figure  and  state  which 
of  the  three  lower  lines  is  AB  prolonged. 
Then  test  your  answer  by  laying  a  ruler 
along  AB. 

(g)  In  the  figures  below,  are  the  lines 
everywhere  the  same  distance  apart?  Test  your  answer  by  using 
a  ruler  or  a  slip  of  paper. 


\   \    \   \   \   \   \   \ 
\   \    \   \   \    \   \  \ 


(h)  In  the  diagrams  given  below,  tell  which'  sect  of  each  pair  is 
the  longer,  a  or  6,  and  test  your  answer  by  careful  measurement. 


{Z 


(i)  In  the  figures    below,  tell    which  lines  are    prolongations 
of  other  lines.    Test   your  s  /f 

answers.  i ^~i 

104.   (a)    Draw  two  un-  X 

\  /  [ 

equal  supplementary-adja-    L~7 — 
cent  angles.  * 

(b)  Extend  the  common  side  of  these  angles  through  the  vertex, 
and  call  the  angles  thus  formed   a,  &  y,   d. 


INTRODUCTION 


35 


(c)  What  relation  exists  between  a  and  0? 

(d)  What  relation  exists  between  0  and  7? 

0)  What  further  relation  do  you  notice  that  is  based  upon  the 
relatons  stated  in  (c)  and  (d)? 

(/)  Do  the  same,  using  angles  0,  7,  5- 
(0)  Do  the  same,  using  angles  7,  6,  a. 
(&)  What  conclusion  can  you  draw? 
105.  If  a  plumb-line  is  fastened  to  a 
horizontal  wire  nail  at  the  vertex  of  the 
angle  of  a  quadrant,  and  the  quadrant 
is  turned  so  that  the    plumb-line   falls 
along  90°   (here  indicated  by  OA),    by 
noting  where  the  shadow    of    the    nail 
strikes  the   quadrant  the   angular   alti- 
tude of  the  sun  may  be  obtained.     Ex- 
plain why  OC  in  the  diagram  gives  the 
angular  altitude  of  the  sun. 

INSTRUMENTS  FOR  MEASURING  ANGLES 

Surveyors  and  engineers  em- 
ploy for  measuring  angles  costly 
instruments  called  theodolites.* 
An  inexpensive  substitute  for  a 
theodolite  is  shown  in  the  accom- 
panying figure  1.  It  consists  of 

*  The  identical  theodolite  with  which  the  historic  M  ason-Dixon  line,  be- 
tween Maryland  and  Pennsylvania  was  run,  settling  a  controversy  of  a  century 
growing  out  of  the  overlapping  charters  of  Charles  I  to  Lord  Baltimore  and 
Charles  II  to  William  Penn,  has  lately  become  the  possession  of  the  Royal  Geo- 
graphical Society,  of  London,  through  Edward  Dixon,  descendant  of  Jeremiah 
Dixon,  who  used  it.  Mason  and  Dixon  had  observed,  for  the  Royal  Society, 
the  transit  of  Venus  at  the  Cape  of  Good  Hope  in  1761,  and  did  their  American 
work  two  years  later.  When  the  line  was  resurveyed  150  years  later,  by  the 
Coast  and  Geodetic  Office  of  Washington,  it  was  proved  to  be  exceptionally 
accurate,  with  no  errors  of  latitude  of  more  than  two  or  three  seconds — cer- 
tainly a  creditable  result  for  the  time  and  the  primitive  instrument  with  which 
the  work  was  done.  The  Mason-Dixon  theodolite  has  two  sights,  a  large  com- 
pass in  the  center  of  its  horizontal  plate,  and  is  adapted  for  measuring  either 
horizontal  circles  or  magnetic  bearings.  The  graduated  circle  is  twelve  inches 
in  diameter,  divided  into  five  minutes,  and  read  by  a  single  vernier. 


B 


FIG.  1 


36 


PLANE  GEOMETRY 


FIG.  2 


two  pieces  of  wood  shaped  like  rulers  mounted  on  a  vertical  axis, 

by  a  pin  driven  through  their  exact  centers.    The  vertical  needles 

inserted  near  the  end  of  the 
rulers  are  used  for  sighting. 
In  place  of  the  needle  near- 
est the  eye,  it  is  better  to 
employ  a  thin  strip  of  wood, 
A,  having  a  fine  vertical  slit; 
and  in  place  of  the  other 
needle,  a  vertical  wire  fixed 
in  a  light  frame,  B.  By  the 

help  of  this  instrument,  and  a  protractor,  one  can  measure  with 

considerable  accuracy  an  angle  on  the  ground;  for  instance,  the 

angle  MON  (figure  2) .    The  following  is 

a  simple  substitute  for   the  theodolite. 

By  means  of  it  angles  may  be  measured 

in  both  horizontal  and  vertical  planes. 

The  vertical  rod  MM'  is  free  to  revolve 

in  the  socket  at  M ,  carrying  a  horizontal 

pointer  which  indicates  readings  on  the 

horizontal   circle  divided   into    degrees. 

These  divisions  must  be  marked.    The 

pointer  at   M'  is  provided  with  sights 

and  is  free  to  move  in  a  vertical  circle 

around  M1 '.    By  sighting  along  this  pointer,  vertical  angles  may 

be  measured  on  the  quadrant. 

(This  instrument  was  suggested  in  Betz  and  Webb,  Plan  e  Geometry .) 

EXERCISES.     SET  XI.     INSTRUMENTS  FOR  MEASURING  ANGLES 

106.  Construct  an  instrument  such  as  that  shown  in  figure  1  of 

the  precedin  g  section  or  an  astrolabe  or  a  good  substitute  by  means  of 

which  angles  may  be  measured  in  vertical  and  horizontal  planes. 

F.  THE  DISCOVERY  OF  SOME  FACTS  AND  THEIR 
INFORMAL  PROOF 

A  theorem  is  the  statement  of  a  fact  which  is  to  be  proved. 

The  fact  which  you  discovered  if  you  worked  exercise  104  and 
applied  in  105  is  one  which  was  known  to  Thales  about  600  B.C. 


INTRODUCTION  37 

It  is  very  important,  so  we  shall  attempt  to  prove  it  again.    It  is 
the  first  theorem  in  our  syllabus. 

Theorem  1.    Vertical  angles  are  equal. 

In  the  accompanying  .diagram,  what  angle  is  the  supplement 
of  ^BOZl 

What  other  angle  is  the  sup- 
plement of  <££OZ? 

What  postulate  can  you 
quote  to  prove  the  equality  of 
these  two  angles  which  are  the 
supplements  of  ^BOZ? 

In  similar  fashion  prove  that  ^BOZ=  <£ AOY. 

A  plane  polygon  is  a  portion  of  a  plane  whose  boundaries  are 
straight  lines.  The  lines  which  bound  the  polygon  are  called  its  sides ; 
the  intersections  of  its  sides  are  the  vertices  of  the  polygon;  and  the 
angles  formed  by  its  sides,  its  angles. 

A  triangle  is  a  polygon  of  three  sides.  Triangles  may  be  classified 
in  at  least  two  ways. 

1.  CLASSIFICATION  BASED  UPON  SIDES 

A  scalene  triangle  is  one  having  no  two  sides  equal. 

An  isosceles  triangle  is  one  having  two  equal  sides. 

An  equilateral  triangle  is  one  having  all  three  sides  equal. 

2.  CLASSIFICATION  BASED  UPON  ANGLES 

An  acute  triangle  is  one  in  which  all  the  angles  are  acute. 

A  right  triangle  is  one  which  has  one  of  its  angles  a  right  angle. 

An  obtuse  triangle  is  one  which  has  one  of  its  angles  an  obtuse  angle. 

The  following  experiment  will  enable  us  to  solve  such  problems 
as  the  one  in  Ex.  110,  in  which  we  would  like  to  find  the  distance 
between  two  points  separated  by  marshland,  but  for  which  we 
lack  the  necessary  information. 

EXPERIMENT 

(a)     Construct  a  triangle  c 

having  two  of  its  sides  equal 
to  sects  b  and  c,  andtheangle 
whose  sides  they  form  equal  to 
Construct  a  second  such  triangle. 


38  PLANE  GEOMETRY 

(6)  Cut  out  and  tear  off  part  of  one  triangle  as  indicated  by  the 
ragged  dotted  line  between  Q  and  P  in  the  diagram,  and  place  it 

upon  the  other  so  that  the  equal 
angles  coincide,  and  so  that  the 
equal  sides  61*  and  62  fall  along 
each  other. 

(c)  Describe  what  happens. 

(d)  What  parts  of  these  tri- 
angles are  you  sure  will  fit  on 
each  other  to  start  with? 

(e)  Where  did   P  fall    and 
where  did  Q  fall  when  you  had 
placed  them  as  you  were  told  to? 

(/)  What  is  it  that  finally  de- 
termines that  the  triangles  may  be  made  to  coincide  throughout? 

(g)  Can  you  quote  a  postulate  to  uphold  your  statement  in  (/)? 

(h)  Repeat  the  process  of  constructing,  cutting,  and  placing 
when  the  given  <£A  is  a  right  angle. 

(i)  Repeat  again  when  the  given  <£A  is  an  obtuse  angle. 

(j)  What  conclusion  can  you  draw? 

Such  triangles  as  AiXY  and  A2PQ  in  the  figure, which  may  be 
made  to  coincide  throughout  are  said  to  be  congruent. 

Congruent  polygons  are  those  which  may  be  made  to  coincide 
throughout.  Hence  they  have  not  only  the  same  shape  but  the 
same  size.  The  symbol  used  to  denote  congruence  is  ^ ,  which  is 
simply  the  sign  of  equality  in  addition  to  the  initial  letter  of  the 
word  similis  (Latin  for  similar)  thrown  down  on  its  side.  • 

Superposition  Postulate.  It  may  be  noted  that  in  the  preceding 
experiment  we  have  assumed  the  following  fact.  Any  geometric 
figure  may  be  moved  about  without  changing  its  size  or  shape. 

EXERCISES.    SET  XII.    MEANING  OF  CONGRUENCE  AND 
CLASSIFICATION  OF  TRIANGLES 

107.  Draw  a  polygon  having    , 16 

the  same  area  as  the  accom- 
panying figure,  but  not  the  same 
shape. 


90° 


*Read  "b  sub-one,"  "b  sub-two,"  indicating  that  these  sects  equal  the 
original  sect  b. 


INTRODUCTION 


39 


108.  Draw  a  polygon  having  the  same  shape  as  that  in  exercise 
107,  but  one-fourth  its  area. 

109.  Classify  triangles  1  through 
18  in  the  following  diagram  (a), 
according  to  sides  and  (6)  accord- 
ing to  angles. 

Just  as  no  matter  how  many 
straight  lines  are  drawn  between 
two  points  they  will  all  coincide 
with  one  another,  so,  no  matter 
how  many  triangles  are  constructed  with  two  sides  and  the  angle 
between  them,  equal  each  to  each,  they  may  be  made  to  coincide. 
We  said  that  two  points  determine  a  straight  line.  Likewise 
we  can  state  the  second  theorem  of  our  syllabus  as  follows : 

*  Theorem  2.    Two  sides  and  the  included  angle  determine  a 
triangle. 

EXERCISE.     SET  XIII.      APPLICATION  OF  CONGRUENCE  OF 

TRIANGLES 

110.  Show  how  the  distance 
from  A  to  B  can  be  found  when 
because  of  some  obstruction  it 
cannot  be  measured  directly. 
Suppose  a  commander  of  an 
army  wished  to  find  the  dis- 
tance across  a  stream.  He  would 
have  a  problem  different  from 

that  in  Ex.  110,  and  hence  that  case  in  the  congruence  of  triangles 
wouldn't  help  him.  The  following  experiment  would  enable  him 
to  solve  the  problem,  however,  as  outlined  in  Ex.  111. 


*  It  is  suggested  that  theorems  marked  in  this  way  be  developed  in  the  class- 
room only.  The  proofs  are  introduced  rather  for  the  sake  of  letting  the  pupils 
realize  that  the  sequence  is  a  perfect  chain  than  as  a  test  of  the  pupil's  power. 
In  order  to  avoid  any  possible  danger  of  memory  work,  the  authors  believe  it 
wiser  to  omit  entirely  the  proof  of  such  propositions  as  appear  to  be  beyond 
the  power  of  the  pupil  to  develop  alone. 


40 


PLANE  GEOMETRY 


A. 


EXPERIMENT 

(a)  Construct  a  figure  having  two  of  its  angles  equal  to  <£  A  and 
•£C  respectively,  and  the  side  common  to  the  two  angles  equal  to  the 

sect  b,  as  in  the  ac- 
companying dia- 
gram. Construct  a 
second  such  figure. 
(b)  Cut  one  of 
themout  and  place 
it  upon  the  other 

—3  so  that  the  equal 

parts  coincide. 

(c)  What  would  happen  if  ~ 
A\P  and  CiQ  were  produced?                        /       \ 

(d)  What  would  happen  if  ^'  ^ 
thelinessimilarlyplacedinthe            ^''  \^ 
other  figure  were  produced?      ^/  \ 

(e)  What    conclusion   can  £ i   &  | ^i-I — 

you  draw? 

(/)  Would  all  triangles  having  two  angles  and  the  side  between 
them  equal  each  to  each  be  congruent? 

(Test  for  cases  where  <£C  is  a  right  angle  or  an  obtuse  angle.) 

If  the  test  is  satisfactory  we  can  state  our  third  theorem  as 
follows: 

"Theorem  3.  Two  angles  and  the  included  side  determine  a 
triangle. 

NOTE. — The  discovery  of  Theorems  2  and  3  is  attributed  to  Thales. 

EXERCISES.    SET  XIII  (continued) 

111.  If  it  is  necessary  for  a  commander  of  an  army  to  know  the 
distance  from  B  to  the  inaccessible  point  C  across  a  stream,  he  may 

B  JHfflL n  find  [i  as  follows:    Run  a  line  BD  at 

right  angles  to  BC.  Prolong  CB.  From 

D  locate  a  point  A  in  the  prolongation 
of  CB  so  that  <£  BDA  =  <£CDB.  Meas- 
ure  AB.  Show  that  ~CB=~AB,  and 
hence  that  #C  may  be  found. 


INTRODUCTION 


41 


112.  To  measure  the  distance  from  B  to  the  inaccessible  point 
F,  run  BD  in  any  convenient  direction.  Locate  C,  the  mid-point 
viBD.  From D,  with  an  in- 
strument, run  DE  so  that 
$CDE=3:FBC,  and  lo- 
cated in  line  with  Cand  F. 
Show  that  BF=ED,  and  ^- 


hence  may  be  determined.  E  D 

Such  problems  as  111  suggest  methods  that  could  be  used  in 
spy  work,  since  only  crude  instruments  need  be  employed. 

113.  To  find  the  distance  AC,  when  C  is  inaccessible,  let  B  be  a 
convenient  point  from  which  A  and  C  are 
visible.    Lay  out  a  triangle  ABCi  making 
<£  3=  <£l_and  <£  4=  «£  2.  Show  that  the  dis- 
tance A  C  may  be  found  by  measuring  A  Ci . 
This  is  a  method  attributed  to  Thales  for 
finding  the  distance  of  a  ship  from  shore. 
^  114.    Thales    of    Miletus    is    said   to 

have  invented  another  way  of  finding  A 

the  distance  of  a  ship  from  shore.  This 
method  may  have  been  as  follows: 

Two  rods,  m  and  n,  are  hinged  to- 
gether at  A.  One  arm  m  is  held 
vertically  while  the  other  n  is  pointed 
at  the  ship  S.  Then  the  .instrument  is 
revolved  about  m  as  an  axis  until  n 
points  at  some  familiar  object  Si  on 
the  shore.  Explain  why  BSi  =  BS- 

dl!5.  Tell 
what  measure- 
ments to  make 
to  obtain  the 
distance  be- 
tween two  in- 
a  c  c  e  s  s  i  ble 
points,  A  and 
B,  in  Figure  1. 

dl!6.  Explain  the  method  suggested  by  the  diagram  in  Figure  2 
for  finding  the  distance  from  S  to  the  inaccessible  point  R. 


Q 


FIG.  2 


42  PLANE  GEOMETRY 

SOME  PROPERTIES  OF  THE  ISOSCELES  TRIANGLE 

Before  we  can  solve  many  more  practical  problems  it  will  be 
necessary  for  us  to  collect  a  number  of  geometric  facts.  This  we 
will  do  first  in  a  simple  fashion — by  experimenting — and  second  by 
actual  proof,  for  though  many  of  us  may  already  believe  them  to 
be  true  we  must  be  ready  to  convince  others. 

EXPERIMENT 

(a)  Construct  an  isosceles  triangle  having  sect  b  as  its  base, 
and  each  of  its  equal  sides  equal  to  sect  a.  Bisect  its  vertex 
angle  (i.e.,  the  angle  included  by  the  equal  sides). 

_ (b)   Cut   the   triangle 

out,  crease  along  the  bi- 

, sector  of  the  vertex  angle 

and  see  what  happens. 

(c)  Are  there  any  congruent  triangles?    If  so  can  you  tell  why 
they  are  congruent? 

(d)  Make  a  list  of  three  other  facts  you  have  thus  discovered 
concerning  the  isosceles  triangle. 

(e)  Test  to  see  if  these  facts  are  all  true  when  the  base  b  is 
greater  than  the  equal  sides. 

By  the  bisectors  of  the  angles  of  a  triangle,  those  sects  of  the 
bisectors  are  indicated  which  are  terminated  by  the  opposite  sides. 

The  facts  discovered  in  the  last  experiment  may  be  stated  as 
corollaries  to  the  fact  that  the  bisector  of  the  vertex  angle  of  an 
isosceles  triangle  divides  it  into  two  congruent  triangles.  We  find 
that  the  parts  of  those  triangles  similarly  placed  with  respect  to 
the  parts  known  to  be  equal  to  begin  with,  are  equal.  Such  parts 
are  called  homologous.  Homologous  parts  of  congruent  polygons 
are  always  equal. 

Theorem  4.  The  bisector  of  the  vertex  angle  of  an  isosceles 
triangle  divides  it  into  congruent  triangles. 

COT.  1.  The  angles  opposite  the  equal  sides  of  a  triangle  are 
equal. 

Cor.  2.  The  bisector  of  the  vertex  angle  of  an  isosceles  tri- 
angle bisects  the  base,  and  is  perpendicular  to  it. 


INTRODUCTION 


43 


SOME  PROPERTIES  OF  THE  EQUILATERAL  TRIANGLE 

When  we  study  the  equilateral  triangle  we  find  more  corollaries 
to  theorem  4. 

EXERCISES.    SET  XIV.    EQUILATERAL  TRIANGLES 

117.  What  fact  concerning  the  angles  of  an  equilateral  triangle 
can  you  base  on  the  fact  concerning  the  angles  opposite  the  equal 
sides  of  an  isosceles  triangle? 

118.  How  would  you  state  a  corollary  concerning  the  equilateral 
triangle  corresponding  to  corollary  2  under  theorem  4? 

119.  Construct  an  equilateral  triangle  and 
bisect  any  two  of  its  angles  as  in  the  accom- 
panying diagram. 

120.  Can  you  prove  triangles  ABD  and 
CBX  congruent? 

121.  In  these  triangles  what  side  of  CBX 
is  homologous  to  AD  in  ABD? 

122.  What  conclusion  can  you  draw  con- 
cerning these  sects? 

123.  Try  to  prove  the  same  fact,  using  triangles  A  CD  and 
CAX. 

124.  Could  you  use  two  other  triangles  to  prove  the  same  fact? 
The  results  of  exercises  117-124  may  be  stated  as  follows: 


Theorem  4. 
Cor.  4. 


Cor.  3.    An  equilateral  triangle  is  equiangular. 


The  bisectors  of  the  angles  of  an  equilateral  triangle 
bisect  the  opposite  sides  and  are  perpendicular  to 
them. 

Cor.  5.    The  bisectors  of  the  angles  of  an  equilateral  triangle 
are  equal. 

The  following  experiment  will  help 
us  to  understand  why  it  is  that  a 
long  span  of  a  bridge  in  which  the 
truss  is  made  with  queen-posts  and 
diagonal  rods,  as  shown  in  the  dia- 
gram, is  sufficiently  supported. 


44  PLANE  GEOMETRY 

EXPERIMENT 

(a)  Using  three  sects  of  different  lengths  construct  two  triangles, 
XYZ  and  XPZ  and  place  them  as  in  the  accompanying  diagram 
so  that  the  longest  sides  c  are  coincident,  and  XY=a=XP  are 
2 next  to  each  other. 


(6)  Draw  YP.  What  kind  of 
triangles  are  AXFP  and  AZFP? 
Why? 

(c)    Use  this  fact  to    prove 


a 


^^^^  (d)  In  (c)  what  axiom  did  you 

x/_  c  .J^Arn   have  to  apply  to  pro  ve 

~~  "  "~ 


parts  do  you 
""  \       ^"  Uz  know   to  be    equal    in    AXFZ 

Vp"  and  AXPZ? 

(/)  What  conclusion  can  you  draw  concerning  these  triangles? 
(g)  Test  to  see  whether  you  could  prove  AXYZ  and  &XPZ 
congruent  by  placing  XY  =  a=XP  (the  shortest  sides)  together. 

(h)  If  you  placed  the  triangles  as  suggested  in  (g)  what  axiom 
would  you  have  to  apply  to  prove  two  angles  of  the  triangles  equal? 
(i)  Is  the  conclusion  you  drew  in  (/)  true  here?  If  so,  we  may  state 
Theorem  5.    A  triangle  is  determined  by  its  sides. 

EXERCISES.    SET  XV.    FURTHER  APPLICATIONS  OF 
CONGRUENCE  OF  TRIANGLES 

125.  Why  is  it  that  the  last  case  in  the  congruence  of  triangles  is  se- 
parated from  the  first  two  cases  by  a  theorem  on  the  isosceles  triangle? 

126.  (a)  Three  iron  rods  are  hinged  at 
the  extremities,  as  shown  in  the  diagram. 
Is  the  figure  rigid?    Why? 

(6)  Four  iron  rods  are  hinged,  as  shown 
in  the  diagram.  Is  the  figure  rigid?  If 
not,  how  many  rods  would  you  add  to 
make  it  rigid,  and  where  would  you  add  them? 

NOTE.    This  experiment  can  be  tried  conveniently  with  the  Mecano  toy. 

127.  How  many  diagonal  braces  are  needed  to  support  a  crane? 
Why? 


INTRODUCTION 


45 


F 


128.  How  many  tie-beams  connecting  each  pair  of  rafters  are 
needed  to  brace  a  two-sided  roof  suf- 
ficiently?   Why? 

129.  Note  how  the  girders  of  the 
bridge   in    the  accompanying   diag- 
ram are  fastened.    Why  cannot  the 
bridge  collapse? 

130.  Why  is  the  long  span  of 
the    bridge    represented    in    the 
diagram  sufficiently  supported? 

131.  Prove  by  means  of  con- 
gruent triangles  that  the  directions  given  on  page  19  for  the  dup- 
lication of  an  angle  are  sound. 

132.  Could  you  have  duplicated   *£AOB  in  that  exercise  if 
OD>OC? 

133.  Prove  the  directions  given  for  bisecting  an  angle  correct. 

134.  Prove  that  the  directions  for  erecting  a  perpendicular  to  a 
line  at  a  given  point  in  it  are  correct.    Can  you  suggest  any  varia- 
tion in  this  construction  such  as  is  pointed  out  in  exercise  132? 

d!35.  Prove  that  the  directions  for  bisecting  a  sect  perpendicu- 
larly are  correct.  Compare  this  construction  with  the  bisection  of 
any  angle. 

136.  Prove  that  the  directions  for  dropping  a  perpendicular  from 

a  point  to  a  line  are  correct. 
d!37.  In  the  sixteenth  cen- 
tury, the  distance  from  A  to 
*^  the  inaccessible  point  B  was 
"~  B  found  by  use  of  an  instrument 
consisting  of  a  vertical  staff 
AC,  to  which  was  attached  a 
horizontal  cross  bar  DE  that  could  be  moved  up  and  down  on  the 
staff-  Sighting  from  C  to  B,  DE  was  lowered  or  raised  until  C,  E 
and  B  were  in  a  straight  line.  Then  the  whole  instrument  was  re- 
volved, and  the  point  F  at  which  the  line  of  sight  CEi  struck  the 
ground  again  was  marked,  and  FA  measured.  Show  that  FA  =  AB- 
(This  exercise  is  taken  with  modifications  from  Stone-Millis, 
Plane  Geometry.) 


46  PLANE  GEOMETRY 

d!38.  Cor.  1,  theorem  4,  is  known  as  the  "Pons  asinorum," 
or  "Bridge  of  Asses."  Its  discovery  is  attributed  to  Thales.  The 
proof  suggested  in  this  exercise,  however, 
is  due  to  Euclid.  He  produced  BA  and 
BC,  the  equal  sides  of  the  triangle  to  D  and 
E,  so  that  BD=BE.  Then  he  proved  (1) 
&BCD^ABAE,C2)  &ACD¥  ACAE,  and  so, 
by  subtracting  *£ACD  from  ^BCD,  and 
<£  CAE  from  <£ BAE  found  <£ BAG  =  <£  £CA . 
Give  the  details  of  the  proof. 

LIST  OF  WORDS  DEFINED  IN  CHAPTER  I 

Solid,  surface,  line,  point;  straight, curved  line,  sect;  plane.  Angle,  vertex, 
sides;  straight,  right,  acute,  obtuse  angles;  complementary,  supplementary 
angles;  adjacent,  vertical  angles;  perpendicular.  Polygon,  vertices,  sides, 
angles;  triangle;  scalene,  isosceles,  equilateral  triangles;  acute,  right,  obtuse 
triangles.  Congruent,  homologous.  Theorem,  corollary,  axiom,  postulate. 

SUMMARY  OF  AXIOMS  IN  CHAPTER  I 

1.  The  sums  of  equals  added  to  equals  are  equal. 

2.  The  remainders  of  equals  subtracted  from  equals  are  equal. 

3.  The  products  of  equals  multiplied  by  equals  are  equal. 

4.  The  quotients  of  equals  divided  by  equals  are  equal. 

5.  A  quantity  may  be  substituted  for  its  equal  in  a  statement  of  equality 
or  inequality. 

6.  Two  quantities  which  are  equal  to  equal  quantities  are  equal  to  each 
other. 

7.  The  whole  is  equal  to  the  sum  of  its  parts. 

SUMMARY  OF  POSTULATES  IN  CHAPTER  I 

Straight  Line 

1.  Two  intersecting  straight  lines  determine  a  point. 

2.  Two  points  determine  a  straight  line. 

3.  A  straight  line  is  the  shortest  distance  between  two  points. 

Angle 

4.  All  straight  angles  are  equal. 

Cor.  1.    All  right  angles  are  equal. 

Cor.  2.    Complements  of  the  same  angle  or  equal  angles  are  equal. 
Cor.  3.    Supplements  of  the  same  angle  or  equal  angles  are  equal. 
Superposition 

5.  Any  geometric  figure  may  be  moved  about  without  changing  its  size  or 
shape. 


INTRODUCTION  47 

SUMMARY  OF  THEOREMS  PROVED  IN  CHAPTER  I 

1.  Vertical  angles  are  equal. 

2.  Triangles  are  determined  by  two  sides  and  the  included  angle. 

3.  Triangles  are  determined  by  two  angles  and  the  included  side. 

4.  The  bisector  of  the  vertex  angle  of  an  isosceles  triangle  divides  the 
triangle  into  two  congruent  triangles. 

Cor.  1.    The  angles  opposite  the  equal  sides  of  a  triangle  are  equal. 
Cor.  2.    The  bisector  of  the  vertex  angle  of  an  isosceles  triangle 

bisects  the  base,  and  is  perpendicular  to  it. 
Cor.  3.    An  equilateral  triangle  is  equiangular. 
Cor.  4.    The  bisectors  of  the  angles  of  an  equilateral  triangle  bisect 

the  opposite  sides  and  are  perpendicular  to  them. 
Cor.  5.    The  bisectors  of  the  angles  of  an  equilateral  triangle  are 

equal.  I 

5.  Triangles  are  determined  by  their  sides. 


CHAPTER  II 


THE  PERPENDICULAR,    THE   RIGHT   TRIANGLE  AND 

PARALLELS 
A.  THE  PERPENDICULAR 

We  have  been  studying  the  congruence  of  triangles  in  general,  and 
as  a  necessary  and  interesting  part  of  that  topic  we  have  considered 
some  properties  of  the  isosceles  triangle.  Now  we  are  to  give  our 
attention  to  another  special  kind,  the  right  triangle,  but  before 
doing  so,  we  need  to  know  more  than  we  do  about  perpendiculars. 

Two  facts  that  we  should  note  at  the  beginning  are  sufficiently 
obvious  to  permit  our  accepting  them  without  proof,  i.e.,  postu- 
lating them. 

Postulates  of  Perpendiculars.  1.  At  a  point  in  a  line  only  one 
perpendicular  can  be  erected  to  that  line. 

2.  JZwmapoint  outside  a  line  only  one  perpendicular  can  be  drawn 
jjine. 
icause  of  this  property  of  perpendiculars,  by  the  distance  from 

point  to  a  line  is  meant  the  length  of  a  perpendicular  from  the 
point  to  the  line. 

TEeTess  we  postulate  and  the  more  we  prove,  the  more  scientific 
is  our  work.  Hence,  later  in  our  study  of  geometry,  we  shall  prove 
these  postulates  of  perpendiculars. 

SET  XVI.    DISTANCE  FROM  A  POINT  TO  A  LINE 

139.  Prove  the  familiar  fact  that 
the  image  of  an  object  in  a  mirror 
appears  to  be  as  far  behind  the  mirror 
as  the  object  is  in  front  of  it. 

Hints:  (a)  It  is  proved  in  physics  that  a 
ray  of  light  striking  a  plane  surface  is  reflected 
from  it  at  the  same  angle  as  it  strikes  it. 
Assume  that  fact  here,  (b)  i,  M,  R  lie  in  a 
straight  line.  See  the  diagram,  (c)  Prove 
triangles  congruent. 

Before  proving  another  important  property  of  perpendiculars 
we  must  add  to  our  list  of  axioms. 
48 


PERPENDICULAR,  RIGHT  TRIANGLE,  PARALLELS    49 

Axioms  of  Inequality.  1.  //  unequals  are  operated  on  by 
positive  equals  in  the  same  way,  the  results  are  unequal  in  the 
same  order. 

In  other  words,  if  we  add  a  positive  number  to  each  of  two 
unequal  numbers  the  sums  will  be  unequal  in  just  the  same  way 
that  the  original  numbers  were  unequal,  i.e.,  the  greater  num- 
ber increased  will  still  be  greater  than  the  smaller  number 
increased;  e.g.,  7>5  and  7+2>5+2.  Would  the  same  be  true 
if  we  started  with  7>  —5,  or  -7<  —5?  Give  your  reason. 

If  instead  of  adding  equal  numbers  to  the  two  unequal  num- 
bers, we  had  subtracted  equal  numbers  from  each,  or  divided 
or  multiplied  each  by  equal  numbers,  the  remainders,  quotients, 
or  products  would  have  been  unequal  in  the  same  order  as  the 
original  numbers. 

2.  //  unequals  are  subtracted  Jrom  equals,  the  remainders  are 
unequal  in  the  reverse  order. 

Example.  10=10 

and  6>3 

.'.  4<7 

EXERCISES.    SET  XVII.    NUMERICAL  INEQUALITY 

140.  State  in  algebraic  symbols  the  above  axioms  of  inequality. 

141.  (a)  Why  is  it  that  if  -2<  -1,  (  -2)2n>(  -I)2"? 
(6)  Why  is  this  not  an  exception  to  our  axiom? 

We  are  now  ready  to  prove  another  important  fact  about  per- 
pendiculars which  will  give  us  another  reason  for  measuring  dis- 
tance from  a  point  to  a  line  by  means  of  the  perpendicular. 

Theorem  6.  The  perpendicular  is  the  shortest  sect  that  can 
be  drawn  from  a  point  to  a  line. 

To  show  that  from  P  a  point  outside 
of  ABj  the  shortest  sect  that  we  can 
draw  to  AB  is  the  perpendicular  PQ, 

let  us  draw  any  other  sect,  say^P/S  to     A Q_ 

AB,  and  then  show  that  PQ  <PS.    So 
far  in  our  study  of  geometry  we  have 
no  statement  which  will  lead  us  to  this 
conclusion  directly. 
4 


50 


PLANE  GEOMETRY 


The  axiom  of  inequality  which  we  ha\>e  just  discussed  might 
suggest,  however,  that  we  can  derive  the  conclusion  we  wish  by 

showing  that  2PQ<2PS. 

Then,  we  also  recall  that  we  ha\>e 
listed  one  fact  concerning  the  inequality 
of  sects  and  that  is  the  postulate  that  a 
straight  line  is  the  shortest  distance  be- 
tween two  points.    Thus  we  are  led  to 
make   the  construction   shown   in   the 
diagram^  i.e.,  produce  PQ  to  R  so  that 
connect  R  and  S. 
Why? 


Finish  the  proof. 


Now 

PS+RS=2PSti  PS=RS. 

Try  to  prove  PS=RS. 

How  do  we  prove  sects  equal? 


EXERCISES.    SET  XVIII.    INEQUALITY  OF  SECTS 

142.  Given  the  point  P  outside  AB  and  L  in  the  line.   Which 
is  shorter,  the  distance  from  P  to  L  or 

the  distance  from  P  to  AB1  -P 

143.  Which  is  the  longest  side  of  a 

right  triangle?  Give  a  reason  for  your  -  :  —  j  --  ^ 
answer. 

B.  THE  RIGHT  TRIANGLE 

We  are  now  ready  to  go  on  with  the  study  of  the  congru- 
ence of  triangles  by  noting  two  cases  of  the  congruence  of  right 
triangles. 

EXPERIMENT 

Construct  a  right  triangle  given  the  side  opposite  the  right  angle, 
called  the  hypotenuse,  and  an  angle  adjacent  to  it.  Let  us  see  if 
we  have  data  sufficient  to  determine  the  triangle. 

(a)  What  is  the  only  side  of  the  triangle  that  is  known? 

(6)  Where  then  will  you  have  to  start  the  construction? 

(c)  Is  the  direction  of  a  second  side  fixed?    Why? 

(d)  Is  the  direction  of  the  third  side  then  fixed?    Why? 
We  are  thus  led  to  expect  : 

*Theorem  7.  The  hypotenuse  and  adjacent  angle  determine 
a  right  triangle. 


PERPENDICULAR,  RIGHT  TRIANGLE,  PARALLELS*   51 


Let  us  convince  ourselves  of  the  truth  or  falsity  of  this  conclusion 
by  actual  proof. 

Given:  A  ABC  and  AXYZ 
with  4^  and  4F  each  a  rt. 
4  and  4C  =  4Z  and  AC  = 
XZ. 

To  prove:  &ABC  =  &XYZ 

Outline  of  proof  :     Place  

AABC  on  &XYZ  so  that  A(?     #_  ~~C 

coincides  with  its  equal  XZ  and  CB  falls  along  ZY. 
do  this? 


Y  Z 

What  right  have  we  to 


Then  AB  will  faU  along  X  Y.    Why?    Finally  B  will  coincide  with  Y.    Why? 
The  other  case  of  congruence  of  right  triangles  is: 
Theorem  8.    The  hypotenuse  and  another  side  determine  a 
right  triangle. 

Given:  AABCand  A^FZwith AB=XY}  AC^XZfmd  4#  and  4Feach 
art. 4. 

Toprove:  AA£C  £  AXYZ.  A 

Suppose  we  place  &XYZ 
next  to  AABC  (as  in  the  fol- 
lowing diagram)  with  XY 
coinciding  with  AB.  (Do 
we  know  that  we  can?) 

(1)  Why  will  the  figure 

formed  be  a  triangle?  ^ 

(2)  What  kind  of  triangle  will  be  formed? 
(3)  Can  you  now  throw  this  theorem  back  to  the  previous  one? 

C.   PARALLELS 

Parallels,  or  parallel  straight  lines,  are  coplanar  lines  (lines  lying 
in  the  same  plane)  which  never  meet.  Do  you  see  any  reason  for 
emphasizing  the  fact  that  the  lines  must  lie  in  the  same  plane? 
Take  two  pencils  and  hold  them  so  that  they  would  neither  meet 
if  continued  nor  be  parallel. 

Draw  a  line  on  a  piece  of  paper  and  erect  two  perpendiculars 
to  it.  Do  these  perpendiculars  appear  to  be  parallel?  Since  they 
lie  in  the  same  plane  they  must  either  be  parallel  or  meet.  Can 
they  meet?  Give  a  reason  for  your  answer.  This  leads  us  to  state 
another  theorem  for  our  syllabus,  one  which  is  frequently  used  in 
mechanical  drawing. 


52 


PLANE  GEOMETRY 


Theorem  9.    Lines  perpendicular  to  the  same  line  are  parallel. 

EXERCISES.    SET  XIX.    PARALLELS 

144.  What  principle  is  a  carpenter  using  when  he  lays  off  parallel 
lines  on  a  board  by  moving  one  arm  of  his  square  along  a  straight 
edge  of  the  board,  and  marking  along  the  other  arm? 

145.  What  is  the  principle  involved  in  the 
use  of  the  T-square  for  drawing  parallels? 
d!46.  The  accompanying  picture  shows 
a  carpenter's  plumb-level,  the  forerunner 
of  the  spirit-level.  AE  and  EB  are  strips 
of  wood  of  equal  length.  CE=ED  and  0 
is  the  midpoint  of  CD.  A  and  B  rest  on 
the  points  to  be  levelled,  and  they  are  found  to  be  level  when  EF 
passes  through  0.  Explain. 

Before  proceeding  with  our  study  of  parallels,  we  need  the 
Postulate  of  Parallels.    Through 
a  given  point  only  one  line  can  be 
drawn  parallel  to  a  givtn  line. 

Not  both  lines  a  and  b  can  be   . 

parallel  to  c.    Why? 

Cor.  1.  Lines  parallel  to  the  same  line  are  parallel  to  one  another. 

Suggestion  for  proof:    If  lines  a,  b,  both  parallel  to  c,  should  meet  how  would 
the  postulate  of  parallels  be  violated? 

*Theorem  10.  A  line  perpendicular  to  one  of  a  series  of  par- 
allels is  perpendicular  to  the  others. 

Given:  AXB  \\  CYD\.  JXYF  _L  AXB. 
To  prove:  EFA.CD. 
Suggestions:  Draw  YK±EF. 

What  relation  will  exist  between  YK  and 
AB1 

Why  will  YK  and  CD  coincide? 

What   relation   exists    between  CD   and 
EF1    Why? 

Why  is  it  necessary  to  consider  only  two 
parallels?   

*  When  naming  a  line  by  more  than  two  of  its  points  it  is  necessary  to  use 
a  bar  over  the  letters.    In  the  case  of  two  points  it  is  immaterial.    Why? 
f  How  do  the  statements  given  show  that  EF  cuts  AB  and  CD? 


C 


D 


PERPENDICULAR,  RIGHT  TRIANGLE,  PARALLELS     53 

EXERCISE.    SET  XIX  (continued) 

147.  Representing  a  series  of  lines  by  p,  q,  r,  .  . .  .  construct  the 
following  figures,  stating  in  each  case  all  possible  relative  positions 
of  the  first  and  last  line  of  the  series : 

(a)  p  ||  q,  p  ||  r.  (b)  p  \\  q,  q  \\  r.  (c)  p±q,  q±r.  (d)  p±q, 
q\\r.  (e)  p_\_q,  q\[r,  r_Ls.  (/)  p  \\  q,  q±r,  r  ||  s. 

ANGLES  FURTHER  DEFINED  ACCORDING  TO  RELATIVE  POSITION 

If,  as  in  the  accompanying  diagram, 
ABC  and  DEF  are  cut  by  GBEH, which 
is  called  a  transversal  (since  it  cuts 
across),  certain  sets  of  angles  are 
formed  to  which  for  brevity  we  give 
the  following  names : 

^ABE  and  ^FEB  are  called  alter- 
nate-interior angles. 

and  ^HEF  are  called  alternate-exterior  angles. 

and  $.FEB  are  called  consecutive-interior  angles. 

and  <$.DEH  are  called  consecutive-exterior  angles. 

and  ^FEB  are  called  corresponding  angles. 


EXERCISES.    SET  XX.    RELATIVE  POSITION  OF  ANGLES 

148.  Only  one  pair  of  each  kind  of  angles  is  mentioned  in  the 
last  paragraph,  though  there  are  two  pairs  of  each  except  the  last 
kind.    Name  the  second  pair  in  each  case,  and  three  remaining 
pairs  of  corresponding  angles. 

149.  Explain  the  meaning  of  alternate  as  used  here. 

150.  Explain  the  meaning  of  consecutive  as  used  here. 

151.  Explain  the  meaning  of  interior  as  used  here. 

152.  Explain  the  meaning  of  exterior  as  used  here. 

153.  What  kind  of  angles  with  regard  to  relative  position  are 
formed  in  the  letter  Z?    In  the  letter  A?    In  the  letter  #?    HI    N1 


54 


PLANE  GEOMETRY 


154.  In  the  accompanying  diagram  select  angles  under  each 
j  class  you  know  (including  adjacent  and 

vertical)  using: 

(a)  c  and  d  with  a  as  transversal. 

(b)  c  and  d  with  b  as  transversal. 

(c)  a  and  b  with  c  as  transversal. 

(d)  a  and  6  with  d  as  transversal. 

PROPERTIES  OF  PARALLELS 

Theorem  11.  If,  when  lines  are  cut  by  a  transversal,  the  alter- 
nate-interior angles  are   equal, 
the  lines  thus  cut  are  parallel.      ^  - 

Given:  ABC,  XYZ  cut  by  BY. 


a 


To  prove:  ABC  \\XYZ. 


F'  # 


Z 


Take   Q   in   BY   so   that   QB 
Draw  QP1.AB  cutting  AB  at  P. 

Extend  PQ  to  R  in  XZ. 
Then  in  APQB  and  ARQY, 

(1)  QB  =  QY 

(2) 

(3) 

(4) 


PROOF 
QY. 


(5) 


(6)  But  PQR±AB 

(7) 
(8) 


(9)  .'.  PQRA.XRZ 

(10)  . 


(1)  Construction. 

(2)  Data. 

(3)  Vertical  angles. 

(4)  Two  angles  and  the  included  side 

equal  each  to  each. 

(5)  Homologous  parts  of  congruent 

triangles. 

(6)  Construction. 

(7)  Definition  of  perpendicular. 

(8)  Quantities  equal  to  the  same  quan- 

tity are  equal  to  each  other. 

(9)  Definition  of  perpendicular. 

(10)  Lines  perpendicular  to  the  same 

line  are  parallel. 

The  student's  attention  is  called  to  the  form  and  arrangement  of 
this  demonstration,  as  it  is  the  first  formal  proof  given  in  the  text. 
Note  that  after  the  general  statement  of  the  theorem  following  the 
words  "given"  and  "to  prove,"*  specific  statements  are  given 

*  In  place  of  the  word  "given"  either  "data"  or  "hypothesis"  is  frequently 
used,  and  in  place  of  "prove"  the  word  "conclusion." 


PERPENDICULAR,  RIGHT  TRIANGLE,  PARALLELS     55 

referring  to  the  particular  diagram  drawn.  These  statements 
should  be  as  brief  as  possible,  and  such,  that  were  the  diagrams 
erased,  it  could  be  reconstructed.  The  steps  of  the  proof  and  the 
reasons  for  them  are  arranged  in  parallel  columns.  The  con- 
venience of  such  an  arrangement  is  at  once  apparent  if  it  be  com- 
pared with  a  proof  written  in  essay  form.  Write  the  proof  that 
way  and  draw  your  own  conclusions  as  to  which  you  would  prefer 
to  use,  giving  your  reasons. 

Show  that  the  following  corollaries  are  true  by  showing  that  a 
pair  of  alternate-interior  angles  are  equal. 
Cor.  1.    If  the  alternate-exterior  angles  or  corresponding  angles 
are  equal  when  lines  are  cut  by  a  transversal,  the  lines 
thus  cut  are  parallel. 

Cor.  2.  If  either  the  consecutive-interior  angles  or  the  con- 
secutive-exterior angles  are  supplementary  when  lines 
are  cut  by  a  transversal,  the  lines  thus  cut  are  parallel. 

EXERCISES.    SET  XXI.    CONSTRUCTION  OF  PARALLELS 

155.  Parallels  may  be  constructed  by  using  a  T-square  and  a 
triangle.    Explain. 

156.  Draw  three  pairs  of  parallel  lines  using  successively  each 
of  the  three  sides  of  one  of  your  triangles  against  a  ruler. 

157.  By  using  your  knowledge  of  corresponding  angles,  draw  a 
line  through  a  given  point,  and  parallel  to  a  given  line. 

158.  (a)  Practice  drawing  parallels  with  compasses  and  ruler 
until  you  can  draw  them  accurately.    (6)  Test  your  work  by  draw- 
ing any  transversal,  and  measuring  a  pair  of  angles  that  should  be 
equal,     (c)  Which  is  more  likely  to  be  in  error,  your  drawing  or 
your  test? 

159.  The   diagram   suggests  a 
method  of  running  one  line  parallel 
to  another  when  you  are  on  a  field 
without    a    transit.     Explain    and 
justify  the  procedure. 

Presently  we  shall  prove  a  theorem  which  is  closely  related  to 
theorem  11.  Before  doing  so,  however,  we  shall  want  to  see  why 
it  need  be  proved  at  all. 


56  PLANE  GEOMETRY 

EXERCISES.    SET  XXII.    RELATED  STATEMENTS 

160.  (a)  Is  it  true  that  if  a  triangle  is  equilateral,  it  is  also 
isosceles? 

(6)  Is  it  true  that  if  a  triangle  is  isosceles,  it  is  also  equilateral? 

161.  (a)  Is  it  true  that  if  two  angles  are  right  angles,  they  are 
equal? 

(6)  Is  it  true  that  if  two  angles  are  equal,  they  are  right  angles? 

162.  (a)  Is  it  true  that  all  men  are  bipeds? 
(6)  Is  it  true  that  all  bipeds  are  men? 

163.  (a)  Is  it  true  that  if  a  man  lives  in  Cincinnati,  he  lives  in 
Ohio? 

(6)  Is  it  true  that  if  a  man  lives  in  Ohio,  he  lives  in  Cincinnati? 

164.  Explain  how  each  of  the  statements  (a)  and  (6)  in  each  of 
the  four  preceding  exercises  is  related  to  the  other:  that  is,  how 
can  (a)  in  each  case  be  formed  from  (6),  and  how  can  (6)  be  formed 
from  (a)? 

165.  Make  a  statement  related  to  each  of  the  following  as  (6) 
is  related  to  (a)  in  each  of  exercises  16Q  to  163,  and  tell  whether 
or  not  your  statements  are  true. 

(a)  If  a  man  lives,  he  breathes. 

(6)  If  a  polygon  is  a  triangle,  it  has  three  sides. 

(c)  If  it  rains,  the  ground  is  wet. 

166.  From  exercises  160  to  163  and  165  what  can  you  conclude 
as  to  the  truth  or  falsity  of  two  statements  related  in  this  way? 
(a)  May  both  of  them  be  true?    (6)  May  one  of  thfem  be  true  and 
the  other  false? 

167.  If,  then,  we  have  proved  that  a  statement  is  true,  is  it  neces- 
sary to  prove  a  statement  related  to  it  as  the  second  is  to  the  first 
in  each  of  those  exercises,  or  may  we  take  it  for  granted  that  the 
second  will  be  true  without  proof? 

Statements  related  as  those  in  the  last  set  of  exercises,  are  called 
converse  statements.  We  saw  that  each  of  two  converse  state- 
ments could  be  formed  from  the  other  by  interchanging  the  data  or 
hypotheses  with  the  conclusion  or  conclusions;  that  is,  the  two  state- 
ments were  so  related  that  what  was  given  in  each  was  what  was 
supposed  to  follow  in  the  other.  From  the  fact  that  it  is  so  much 


PERPENDICULAR,  RIGHT  TRIANGLE,  PARALLELS    57 

easier  to  make  a  statement  whose  converse  is  absurd  than  one 
whose  converse  is  true,  it  appears  that  we  should  never  claim  that 
the  converse  of  a  theorem  in  geometry  is  true  without  having 
proved  it  so. 

EXERCISES.    SET  XXII  (continued) 

168.  Make  a  statement  of  something  in  life  which  you  know  to 
be  true,  but  whose  converse  is  false. 

169.  Make  a  statement  of  something  in  life  which  you  know  to 
be  true,  but  whose  converse  is  true. 

170.  Do  the  same  as  you  were  requested  to  do  in  the  last  two 
exercises,  but  take  the  statements  from  geometry. 

171.  Select  from  the  theorems  already  proved  two  that  are 
converse  statements. 

172.  The  converse  of  a  definition  is  always  true.     Test  your 
definitions  of  Chapter  I  from  this  point  of  view.    (See  lists  at  end 
of  Chapter  I,  page  46.) 

173.  State  what  was  given  us  in  theorem  11. 

174.  State  what  we  proved  in  theorem  11. 

175.  State  what  would  be  given  in  the  converse  of  theorem  11. 

176.  State  what  would  have  to  be  proved  in  the  converse  of 
theorem  11. 

Theorem  12.    Parallels  cut  by  a  transversal  form  equal  alter- 
nate-interior angles.  \ 

Fill    in    all    the    blank    spaces  X~ 
and  answer  the  questions  in  the 
following: 

Given:  p  R 

To  prove: 

PROOF 


(1)  If  through  the  mid-point  of 
YQ  we  were  to  draw  a  line  perpendic- 
ular to  XZ  what  other  fact  would  we 
know  about  that  line? 

(2)  What    parts    have    we    then 
equal  in  the  triangles  thus  formed? 

(3)  By   what   method   could   we 
then  prove  the  fact  we  wish  to  prove? 


(1)  Why? 


(2)  How  do  you  know  each   of 
these  pairs  are  so  related? 


58 


PLANE  GEOMETRY 


R 


COT.  1.  Parallels  cut  by  a  transversal  form  equal  corresponding 
and  equal  alternate-exterior  angles. 

Cor.  2.  Parallels  cut  by  a  transversal  form  supplementary  con- 
secutive-interior',  and  supplementary  consecutive-ex- 
terior angles. 

EXERCISES.     SET  XXIII.     APPLICATIONS  OF  PARALLELISM 

177.  The  accompanying  diagram 
suggests  a  convenient  method  of 
measuring  the  distance  from  B  to 
an  inaccessible  point  F.  Explain. 

Note  DE  ||  FB. 

How  can  this  be  done  on  the  ground? 

178.  The  " square  network"  shown  in  the  figure  is  used  in 
designing  for  drawing  a  great  variety  of  patterns.  The  patterns 
A  and  B  drawn  upon  it  are  examples  of  Arabian  frets.  The  best 
way  to  rule  the  square  network  is  to 
draw  a  horizontal  line  MN  and  mark 
off  equal  divisions  on  it.  At  each 
point  of  division,  by  use  of  the  tri- 
angle, draw  two  lines,  such  as  PQ 
and  PR,  each  making  an  angle  of  45° 
with  MN. 

Draw  such  a  network,  then  upon  it  construct  a  pattern,  either 
an  original  design  or  a  copy  of  these  Arabian  frets. 
(Taken  from  Stone-Millis,  Plane  Geometry.) 

179.  In  the  annexed  dia- 
gram if  XY\\PQzudRS\\  VT 
how  many  angles  would  you 
need  to  know  in  order  to  find 
the  remaining  angles? 

180.  If  the  side  AC  of   a 
triangle    ABC    is   extended, 
as  in  the  annexed  diagram, 
how  could  a  line  be  drawn 

~~b  through  C  to  make  an  angle 
Would  any  other  angles  be  equal?     Why? 


A 
equal  to  the 


PERPENDICULAR,  RIGHT  TRIANGLE,  PARALLELS    59 

LIST  OF  WORDS  DEFINED  IN  CHAPTER  II 

Distance  (point  to  a  line).  Hypotenuse.  Parallels,  coplanar,  transversal; 
alternate-interior,  alternate-exterior,  consecutive-interior,  consecutive-ex- 
terior, corresponding  angles.  Converse. 

SUMMARY  OF  AXIOMS  IN  CHAPTER  H 

Inequality 

1.  If  unequals  are  operated  upon  by  positive  equals  in  the  same  way,  the 
results  are  unequal  in  the  same  order. 

2.  If  unequals  are  subtracted  from  equals,  the  remainders  are  unequal  in 
the  reverse  order. 

SUMMARY  OF  POSTULATES  IN  CHAPTER  H 

Perpendiculars 

1.  At  a  point  in  a  line  only  one  perpendicular  can  be  erected  to  that  line. 

2.  From  a  point  outside  a  line  only  one  perpendicular  can  be  drawn  to 
that  line. 

Parallels 

3.  Through  a  given  point  only  one  line  can  be  drawn  parallel  to  a  given 
line. 

Cor.  1.    Lines  parallel  to  the  same  line  are  parallel  to  one  another. 

SUMMARY  OF  THEOREMS  IN  CHAPTER  H 

6.  A  perpendicular  is  the  shortest  sect  that  can  be  drawn  from  a  point 
to  a  line. 

7.  The  hypotenuse  and  an  adjacent  angle  determine  a  right  triangle. 

8.  The  hypotenuse  and  another  side  determine  a  right  triangle. 

9.  Lines  perpendicular  to  the  same  line  are  parallel. 

10.  A  line  perpendicular  to  one  of  a  series  of  parallels  is  perpendicular  to 
the  others. 

11.  If  when  lines  are  cut  by  a  transversal  the  alternate-interior  angles  are 
equal  the  lines  thus  cut  are  parallel. 

Cor.  1.  If  the  alternate-exterior  angles  or  corresponding  angles  are 
equal  when  lines  are  cut  by  a  transversal,  the  lines  thus 
cut  are  parallel. 

Cor.  2.  If  either  the  consecutive-interior  angles  or  the  consecu- 
tive-exterior angles  are  supplementary  when  lines  are  cut 
by  a  transversal,  the  lines  thus  cut  are  parallel. 

12.  Parallels  cut  by  a  transversal  form  equal  alternate-ulterior  angles. 

Cor.  1.  Parallels  cut  by  a  transversal  form  equal  corresponding, 
and  equal  alternate-exterior  angles. 

Cor.  2.  Parallels  cut  by  a  transversal  form  supplementary  consecu- 
tive interior,  and  supplementary  consecutive-exterior 
angles. 


CHAPTER  III 

ANGLES  OF  POLYGONS  AND  PROPERTIES  OF 
PARALLELOGRAMS 

A.  ANGLES  OF  POLYGONS 

In  attempting  to  develop  a  formula  for  the  sum  of  the  angles  of 
a  polygon,  it  is  best  for  us  to  begin  with  the  simplest  polygon, 
the  triangle. 

EXERCISE.    SET  XXIV.    SUM  OF  ANGLES  OF  A  TRIANGLE. 

181.  By  reproducing  the  angles  of  a  given  triangle,  place  these 
angles  adjacent  to  one  another.  What  does  the  sum  of  the  angles 
appear  to  be  in  this  case? 

Theorem  13.  The  sum  of  the  angles  of  a  triangle  is  a  straight  angle. 

Prove  this  proposition,  using  the  hints  given  by  the  following 

diagrajn  and  notes: 

Produce  CA  and  draw 
AP  ||  CB. 

What  is  the  sum  of 
A  «£•!,  2,  and  3? 

WTiat  is  the  relation  of  ^2  to  <2i?    Of  ^3  to  $33 

Cor.  1.    A  triangle  can  have  but  one  right  or  one  obtuse  angle. 

Cor.  2.  Triangles  having  two  angles  mutually  equal  are  mutu- 
ally equiangular. 

Cor.  3.  A  triangle  is  determined  by  a  side  and  any  two  homolo- 
gous angles. 

An  exterior  angle  of  a  polyyon  is  one  formed  by  a  side  of  the 
polygon  and  the  prolongation  of  an  adjacent  side. 

In  the  preceding  diagram  <£  RAB  is  an  exterior  angle  of  the  A  ABC. 

Cor.  4.  An  exterior  angle  of  a  triangle  is  equal  to  the  sum  of 
the  non-adjacent  interior  angles. 

60 


POLYGONS  AND  PARALLELOGRAMS      61 

EXERCISES.    SET  XXIV  (continued) 

182.  (a)  The  theorem  that  the  sum  of  the  angles  of  a  triangle 
equals  a  straight  angle  may  be  proved  by  drawing  a  line  through 
a  vertex  parallel  to  the  opposite 
side.  Give  proof. 

NOTE. — This  proof  is  attributed  to 
the  Pythagoreans. 

1  ' "*• —  n 

(b)    Prove  the  same  fact  by 
drawing  a  sect  from  a  vertex  parallel  to  the  opposite  side. 

(c)  Prove   the   same   theorem   by 
drawing  through  any  point  on  one 

^ ^^       side   lines 

A  ^      parallel  to 


the  other  sides  of  the  triangle. 

183.  Prove  this  same  fact  another  A 

way  by  erecting  perpendiculars  to  one  side  at  its  extremities  and 
dropping  a  perpendicular  to  the  same  side  from  the  opposite  vertex. 

184.  Show  how  the  following  procedure  may  be  used  to  test  accu- 
racy with  which  you  measure  angles  with  an  instrument.    Select  the 
three  positions  not  in  a  straight  line;  call  them  stations  A,  B,  and 
C.    From  station  A  measure  the  angle  between  the  directions  to 
B  and  C;  at  B,  measure  the  angle  between  the  directions  to  C 
and  A;  at  C,  the  angle  between  the  directions  to  A  and  B.    Would 
your  measurements  be  accurate,  and  if  not,  what  error  would 
there  be,  if  you  found  the  angles  to  be  respectively:  100°  27',  23°  13', 
and  56°  237 

185.  If  one  angle  of  an  isosceles  triangle  is  60°,  find  the  other 
angles. 

186.  (a)  If  the  vertex  angle  of  an  isosceles  triangle  is  v°  write 
an  expression  for  each  of  the  other  angles. 

(b)  If  a  base  angle  of  an  isosceles  triangle  is  6°  write  an  expres- 
sion for  each  of  the  other  angles. 

187.  What  angle  do  the  bisectors  of  the  acute  angles  of  a  right 
triangle  form? 

188.  Construct  the  following  angles:  60°,300,1200,750,150°,1050. 
The  pupil  is  again  reminded  that  the  use  of  instruments  is 


62 


PLANE  GEOMETRY 


restricted  to  the  straight  edge  and  the  compasses  in  scientific  geo- 
metric constructions. 

d!89.    (a)   Two  mirrors,  mi  and  ra2, 
are  set  so  as  to  form  an  acute  angle  with 
each  other.    A  ray  of  light  is  reflected 
by  mi  so  as  to  strike  m2.    The  ray  is 
again  reflected  by  m2  and  crosses  its 
first  path.   Prove  that  <£rr2= 2  <£mim2. 
(6)  How  should  these  mirrors  be  placed  in  order  that  the  ray 
in  the  second  case  may  be  parallel  to  the  original  ray? 

(c)  Part  (a)  of  this  exercise  is  the  principle  underlying  several 
important  optical  instruments  such  as  the  " optical  square'  and 
the  " sextant."  A  description  of 
them  may  be  found  in  -Any  good 

encyclopaedia    or    in    Gillespie's  ^ 

B 


B 


Surveying,  p.  61,  and  Stone-Millis'  Plane  Geometry,  p.  40,  Ex.  14. 
Look  them  up  and  make  a  crude  optical  square  either  of  paste- 
board or  of  wood. 

A  diagonal  of  a  polygon  is  a  sect  connecting  any  two  non-consecutive 
vertices. 

EXERCISES.    SET  XXV.    SUMS  OF  ANGLES  OF  POLYGONS 

190.  Find  the  sum  of  the  angles  of  a  quadrilateral.    (Can  you  so 
divide  it  into  triangles  that  the  sum  of  the  angles  of  the  triangles 
formed  will  be  the  sum  of  the  angles  of  the  quadrilateral?) 

191.  Draw  a  five-sided  polygon. 

(a)  How  many  diagonals  can  be  drawn  from  one  vertex? 

(6)  How  many  triangles  are  formed  by  drawing  these  diagonals? 


POLYGONS  AND  PARALLELOGRAMS  63 

(c)  What  is  the  sum  of  the  angles  of  the  triangles  thus  formed? 

(d)  What  is  the  sum  of  the  angles  of  a  5-gon? 

Give  a  reason  for  each  of  your  answers.  Check  your  conclusion 
concerning  the  sum  of  the  angles  by  the  use  of  the  protractor. 

A  polygon  is  said  to  be  convex  if  each  of  its  angles  is  less  than  a 
straight  angle.  Only  convex  polygons  will  be  considered  in  the  early 
study  of  geometry.  If  a  polygon  has  four  sides  it  is  called  a  quad- 
rilateral ;  if  five  sides,  a  pentagon ;  six  sides,  a  hexagon ;  seven  sides 
a  heptagon;  eight  sides,  an  octagon;  nine  sides,  a  nonagon;  ten 
sides,  a  decagon;  etc. 

Theorem  14.  The  sum  of  the  angles  of  a  polygon  is  equal  to  a 
straight  angle  taken  as  many  times  less  two  as  the  polygon  has 
sides. 

(1)  Draw  all  the  diagonals  possible  from  one  vertex  of  the 
polygon. 

(2)  If  the  figure  has  n  sides  how  many  triangles  will  you  form 
by  drawing  these  diagonals? 

Give  the  proof  following  the  suggestions  given  in  (1)  and  (2). 
The  pupil  will  find  the  following  (1)  and  (2)  also  give  hints 
leading  to  an  equally  desirable  proof. 

(1)  Connect  any  point  inside  the  polygon  with  each  vertex. 

(2)  If  the  figure  has  n  sides  how  many  triangles  will  you  thus  form? 
A  regular  polygon  is  one  which  is  both  equilateral  and  equiangular. 
Can  you  draw  a  polygon  which  is  equilateral  and  not  equi- 
angular? 

.    Can    you    draw    a    polygon    which    is    equiangular    and  not 
equilateral? 

Cor.  1.    Each  angle  of  a  regular  polygon  of  n  sides  equals  the 
~^th  part  of  a  straight  angle. 

If  the  sides  of  a  polygon  are  produced  in  turn  forming  one  exterior 
angle  at  each  vertex,  these  angles  are  called  the  exterior  angles  of 
the  polygon. 

Could  two  exterior  angles  be  formed  at  each  vertex?  How  would 
two  such  angles  be  related? 


64 


PLANE  GEOMETRY 


Cor.  2.*    The  sum  of  the  exterior  angles  of  a  polygon  is  two 
straight  angles. 

(1)  What  would  be  the  sum  of  the  adjacent  interior  and  exterior 
angle  at  each  vertex? 

(2)  What  would  be  the  sum  of  all  the  interior  and  exterior  angles 
together? 

Cor.  3.    Each  exterior  angle  of  a  regular  polygon  of  n  sides  is 
equal  to  the  —th  part  of  a  straight  angle. 

EXERCISES.    SET  XXV  (continued) 
192.  Fill  in  the  blank  spaces  in  the  accompanying  table: 


No.  of  sides  of 
polygon 

No.  deg.  in  sum 
of  int.  angles 

No.  deg.ineach 
angle  of  regular 
n-gon 

3 

4 

5 

6 

7 

8 

9 

10 

193.  How  many  sides  has  a  regular  polygon  if  each  angle  is 
(a)  150°,  (6)  144°,  (c)  170°,  (d)  175°? 

Hints  on  solution :    ~  (180)  =  150;  or  f  (180)  -  30.    Solve  for  n. 

194.  In  surveying  an  hexagonal  field  the  angles  were  found  to  be 
118°,  124°,  116°^  129°,  130°,  112°. 

(a)  What  error  was  made? 

(6)  Before  making  a  drawing  of  the  survey,  the  engineer  has  to 
distribute  this  error  proportionately  over  all  the  angles  so  as  to 
increase  or  decrease  all  the  angles  according  as  the  sum  of  the  angles 
measured  was  too  small  or  too  great.  Distribute  the  error  correctly 
in  this  case. 

*  Theorem  14  and  this  corollary  were  proved  in  their  general  form  by 
Regiomontanus  (1436-1476),  although  the  facts  were  known  to  earlier  mathe- 
maticians, and  were  proved  by  them  for  special  cases. 


POLYGONS  AND  PARALLELOGRAMS      65 

195.  In  surveying  a  pentagonal  field,  the  angles  were  found  to  be 
A  =  103°  15/,B  =  110°37/,C  =  99°45/,Z)  =  122040/,and  #=102°  15'. 

(a)  What  error  was  made? 

(6)  Distribute  the  error  proportionately. 

196.  (a)  Make  a  list  of  regular  polygons  of  the  same  number  of 
sides  that  may  be  used  to  cover  a  plane  surface  with  a  geometric 
design. 

(6)  For  what  purposes  are  such  designs  used? 

(c)  What  combinations  of  regular  polygons  have  you  seen  used? 

(d)  Sketch  some  of  these  designs. 

(e)  Why  is  it  these  combinations  are  possible? 

197.  Make  as  many  constructions  as  possible  of  each  of  the 
polygons  mentioned,  using  only  rule  and  compasses  for  the  purpose. 

198.  Construct  accurately  a  design  based  upon  each  kind  of 
regular  polygon  or  a  combination  of  regular  polygons  mentioned 
in  Ex.  "196. 

d!99.  Mabel  Sykes,  Source  Book  of  Problems  for  Geometry, 
p.  16,  par.  23,  Ex.  2. 

d200.  Ibid.,  p.  16,  par.  23,  Ex.  4. 

d201.  Ibid.,  pp.  16-17,  par.  24,  Ex.  1. 

d202.  Ibid.,  pp.  16-17,  par.  24,  Ex.  4. 

203.  Prove  the  proposition  concerning  the  sum  of  the  angles 
of  a  polygon,  according  to  the  following  suggestions: 

(a)  By  connecting  a  point  on  one  of  the  sides  of  the  polygon 
with  each  vertex. 

(6)  By  connecting  a  point  outside  of  the  polygon  with  each  vertex. 

The  converse  of  a  proposition  concerning  isosceles  triangles  has 
many  interesting  and  important  applications,  and  we  are  now  ready 
to  prove  it. 

Theorem  15.  //  two  angles  of  a  triangle  are  equal,  the  sides 
opposite  them  are  equal. 

Hint:  Can  you  draw  a  line  to  cut  &ABC 
into  two  triangles  in  such  a  way  that  the  con- 
struction itself  will  make  an  angle  and  a  side 
equal  respectively  in  the  two  triangles? 

Cor.  1.    Equiangular    triangles    are 

equilateral.  c* 


PLANE  GEOMETRY 


B 


EXERCISES.     Set  XXVI.     SIDES  AND  ANGLES  OF  A  TRIANGLE 

204.  (a)  Show  that  a  person  may  find  the  distance  (AC)  at 
which  he  passes  an  object  A,  when  going  in  the  direction  BC,  if 

he  notes  when  his  course  makes  an  angle  of 
45°  with  the  direction  of  the  object  and 
again  when  it  is  at  right  angles  to  the 
object  taking  account  of  the  distance  (BC) 
which  he  traversed  between  the  observa- 
tions. 

(6)  How  could  this  method  be  applied  to 
the  problem  of  finding  at  what  distance 
from  a  lighthouse  a  ship  passes  it? 

205.  To  ascertain  the  height  of  a  tree  or  of  the  school  building, 
fold  a  piece  of  paper  so  as  to  make  an  angle  of  45°.    Then  walk 
back  from  the  tree  until  the  top  is  seen  at  an  angle  of  45°  with  the 
ground  (being,  therefore,  careful  to 

have  the  base  of  the  triangle  level). 
Then  the  height  AC  will  equal  the 
base  A  B,  since  ABC  is  isosceles.  A 
paper  protractor  may  be  used  for 
the  same  purpose.  Can  you  suggest 
a  better  method  than  that  of  meas- 
uring from  the  ground? 

206.  The  distance  of  a  ship  at  sea  may  be  measured  in  the 
following  manner: 

Make  a  large  isosceles  triangle  out  of  wood,  and  standing  at  T, 

sight  to  the  ship  and  along  the  shore 
on  a  line  TA,  using  the  vertex  angle 
of  the  triangle.  Then  go  along  TA 
until  a  point  P  is  reached,  from  which 
T  and  S  can  be  seen  along  the  sides 
of  a  base  angle  of  the  triangle.  Then 
TP  =  TS.  By  measuring  TB,  BS  can 
be  found. 

207.  Distance  can  easily  be  meas- 
ured by  constructing  a  large  equilateral  triangle  of  heavy  paste- 
board, and  standing  pins  at  the  vertices  for  the  purpose  of  sighting. 


POLYGONS  AND  PARALLELOGRAMS      67 

To  measure  PiC,  stand  at  some  convenient  point  A,  and  sight 
along  APC,  and  also  along  AB.  Then  walk  along  AB  until  a  point 
BI  is  reached,  from  which  BiC  makes  with  BiA  an  angle  of  the  tri- 
angle (60°).  Then  prove  that  AC=ABi.  Also,  since  APi  can 
be  measured,  find  P\C. 
C 


\ 


X  A  B  B  BI  ' 

208.  Measure  the  angle  CAX,  either  in  degrees  with  a  pro- 
tractor, or  by  sighting  along  a  piece  of  paper  and  marking  down 
the  angle.  Then  go  along  XA  produced  until  a  point  B  is 
reached  from  which  BC  makes  with  BA  an  angle  equal  to  half  of 
angle  CAX.  Then  show  that  AB=AC.  /L 

NOTE. — A  navigator  uses  the  prin-  /    ' 

ciple  involved  in  the  foregoing  exercises  /'        ' 

when  he  "doubles  the  angle  on  the  bow"  /'  \ 

to  find  his  distance  from  a  lighthouse  or  ^  t**\ 

other  object.    If  he  is   sailing   on   the     £-± '<• — J > 

course  ABC  and  notes  a  lighthouse  L 

when  he  is  at  A,  he  takes  the  angle  A;  and  if  he  notices  when  the  angle  that 
the  lighthouse  makes  with  his  course  is  just  twice  the  angle  noted  at  A ,  then 
BL  =AB.  He  has  AB  from  his  log  (an  instrument  that  tells  how  far  a  ship 
goes  in  a  given  time),  so  he  knows  BL.  He  has  "doubled  the  angle  on  the 
bow"  to  get  this  distance. 

B.     PARALLELOGRAMS 

A  parallelogram  is  a  quadrilateral  whose  opposite  sides  are  parallel. 
Theorem  16.    Either  diagonal  of  a  parallelogram  bisects  it. 

Suggestion:  In  the  proof  note  that  the  diagonal  is  a  transversal  of  the 
parallel  sides. 

Cor.  1.    The  parallel  sides  of  a  parallelogram  are  equal,  and 
the  opposite  angles  are  equal. 

The  sects  of  common  perpendiculars  included  by  parallels  are 
catted  the  distances  between  the  parallels. 

Cor.  2.    Parallels  are  everywhere  equidistant. 


PLANE  GEOMETRY 


EXERCISES.    SET  XXVII.    PARALLELOGRAMS 

209.  One  angle  of  a  parallelogram  is  20°  more  than  three  times 
another.    Find  all  the  angles. 

210.  (a)  Establish  a  relation  between  consecutive  angles  of  a 
parallelogram. 

(b)  How  many  angles  of  a  parallelogram  must  be  known  in 
order  to  determine  the  others? 

211.  A  stairway  inclined  45°  to  the  horizontal  leads  to  a  floor 
15'  above  the  first.    What  is  the  length  of  the  carpet  required  to 

cover  it  if  each  step  is  10"  high?  If  each 
is  12"  high?  If  each  is  9"? 

Can  this  problem  be  solved  without 
knowing  the  height  of  the  steps?  Is  it 
necessary  to  know  that  the  steps  are  of 
the  same  height? 

(Taken  from  Slaught  and  Lennes, 
Plane  Geometry.) 

212.  Is  the  converse  of  the  fact  that  a  diagonal  bisects  a  paral- 
lelogram true  or  false?    Give  a  reason  for  your  answer. 

Theorem  17.    A  quadrilateral  whose  opposite  sides  are  equal 
is  a  parallelogram. 

(1)  What  fact  would  we  need  to  know  about  these  opposite  sides? 

(2)  By  what  method  can  we  obtain  this  fact? 

EXERCISES.    SET  XXVII  (continued) 

213.  An  adjustable  bracket  such  as  dentists  often  use,  is  out- 
lined in  the  figure.    It  is  fastened  to  the  wall  at  A,  and  carries  a 
shelf  B.    Why  is  it  that  as  the  bracket 

is  moved  so  that  B  is  raised  and  low- 
ered, the  shelf  remains  horizontal? 

(Taken    from    Stone-Millis,    Plane 
Geometry.) 

214.  The  accompanying  figure 
is  a  diagram  of  the  "  parallel 
ruler,"  which  is  used  by  designers 
for  drawing  parallel  lines. 


POLYGONS  AND  PARALLELOGRAMS 


69 


(a)  Upon  what  principle  of  parallelograms  must  its  construction 
depend? 

(b)  Make  such  an  instrument. 

A  rectangle  is  a  parallelogram  one  of  whose  angles  is  a  right  angle. 
Cor.    Each  angle  of  a  rectangle  is  a  right  angle.    Why? 

EXERCISE.    SET  XXVII  (continued) 

215.  Prove  that  a  parallelogram  whose  diagonals  are  equal  is  a 
rectangle.    (This  fact  is  used  as  a  check  by  carpenters  and  builders. 
Could  it  be  used  in  laying  out  a  tennis  court?) 

Theorem  18.    A  quadrilateral  having  a  pair  of  sides  both  equal 
and  parallel  is  a  parallelogram. 

EXERCISES.    SET  XXVII  (continued) 

216.  Justify  the  following  method  used  by  surveyors  for  pro- 
longing a  line  beyond  an  obstacle;  that  is,  show  that  in  the  diagram 
EF  is  A  B  prolonged  beyond 

0.  BC  is  run  at  right  angles 
to  AB;  CD±BC;  DE±CD 
and  DE=CB;  EFA.DE. 

217.  The    accompanying 
diagrams  show  another  way 
of  extending  a  line  be- 
yond an  obstacle,    (a) 

By  reference  to  dia- 
gram, state  the  proce- 
dure in  words,  (b)  Show 
that  it  is  correct,  (c)  Compare  this  method  with  that  of  Ex. 
216.  Note,  for  example,  why  two  lines  (CB  and  DE)  are  used 
in  Ex.  217,  and  only  one  (CB)  in  Ex.  216.  Under  what  con- 
D  H  „  ditions  would  you  use  each  method? 

218.  If  the  vertices  of  one  paral- 
lelogram are  in  the  sides  of  another, 
the  diagonals  of  the  two  parallelo- 
grams pass  through  the  same  point 
(called  the  center  of  the  parallelograms) . 

Suggestions:  Call  the  intersection  of  the  diagonals  AC  and  BD,  0.    Draw 
OE,  OF,  OG,  OH  and  prove  EOG  and  FOH  straight  lines. 


E 


70 


PLANE  GEOMETRY 


Pi 


d219.  An  interesting  outdoor  application  of  the  theory  of  paral- 
lelograms is  the  folio  whig:  Suppose  that  you  are  on  the  side  of 
this  stream  opposite  to  XY,  and  wish  to  measure  the  length  of 
XY.  Run  a  line  AB  along  the  bank.  Then  take  a  carpenter's 

square,  or  even  a  large  book,  and 
walk  along  A  B  until  you  reach 
P,  a  point  from  which  you  can 
just  see  X  and  B  along  two  sides 

4      ^ ^^ — B  of  the  square.    Do  the  same  for 

F,  thus  fixing  P  and  Q.  Using 
the  tape,  bisect  PQ  at  M.  Then 
walk  along  YM  produced  until 
you  reach  a  point  FI  that  is  ex- 
actly in  line  with  M  and  F,  and  also  with  P  and  X.  Then  walk 
along  XM  produced  until  you  reach  a  point  Xi  that  is  exactly  in 
line  with  M  and  X}  and  also  with  Q  and  F.  Then  measure 
YiXi  and  you  have  the  length  of  XY.  For  since  YXi_LPQ, 
and  XYi±PQ,  YXi  \\XYi.  And  since  PM  =  MQ,  therefore 
XM=MXl  and  FjM=MF.  Therefore  Y^YX  is  a  parallelo- 
gram. Give  the  reason  for  each  of  these  steps. 

*  Theorem  19.  A  parallelogram  is  determined  by  two  adjacent 
sides  and  an  angle,  or  two  parallelograms  are  congruent  if  two 
adjacent  sides  and  an  angle  are  equal  each  to  each. 

Methods  of  proof. 

(1)  Congruence  of  triangles,  or 

(2)  Superposition,  or 

(3)  Properties  of  parallelograms. 

NOTE. — If  (1)  or  (3)  is  used  it  must  be  shown  that  the  parts  proved  equal 
are  arranged  in  the  same  order  in  the  two  parallelograms. 

EXERCISES.    SET  XXVII  (continued) 

220.  Construct  a  parallelogram,  given 

(a)  Two  adjacent  sides  and  an  included  angle. 
(6)  Two  adjacent  sides  and  a  non-included  angle, 
(c)  A  side,  a  diagonal,  and  the  angle  between  them. 

221.  In  physics  it  is  shown  that  if  two  forces  (such  as  a  push 
and  a  pull)  are  exerted  in  different  directions  upon  the  same  object, 
they  have  the  same  effect  as  a  single  force  called  their  resultant. 


POLYGONS  AND  PARALLELOGRAMS  71 

How  is  this  fact  illustrated  by  the  bean-shooter? 

Referring  to  the  accompanying  diagram,  if  the  directions  and 
magnitudes  of  two  forces  working  on  object  A  are  represented  by 
the  sects  AB  and  AC,  the  direction 
and  magnitude  of  the  resultant  will 
be  represented  by  the  sect  AD,  which 
is  the  diagonal  of  the  parallelogram, 
with  A B  and  AC  as  adjacent  sides. 
In  physics,  such  a  diagram  is,  for 
obvious  reasons,  called  the  Paral- 
lelogram of  Forces. 

A  force  of  50  pounds  is  exerted  upon  a  body  pulling  it  in  one 
direction,  and  at  the  same  time  another  force  of  100  Ibs.  pulls  it 
in  a  direction  at  an  angle  of  45°  with  the  first.  Show  by  the 
parallelogram  of  forces  the  effect  on  the  body. 

NOTE:  (a)  Use  only  compasses  and  ruler  in  solving.  Represent  50  Ibs. 
by  any  given  sect  as  unit;  draw  the  forces  to  scale,  and  find  the  resultant. 

(6)   Use  protractor  and  marked  edge  in  reading  result. 

(c)  Could  you  read  the  resultant  to  any  degree  of  accuracy  by  any 
other  method? 

222.  Two  forces,  250  Ibs.  and  400  Ibs.  respectively,  are  exerted 
upon  a  body  at  right  angles  with  each  other.    Find  their  resultant 
as  in  Ex.  221.    Check  the  result  by  computation.    Why  was  such 
a  check  not  available  for  you  in  Ex.  221? 

223.  Find  the  resultant  of  two  forces  exerted  upon  a  body  at 
an  angle  of  150°  with  each  other,  one  of  50  Ibs.,  the  other  of  60  Ibs. 

224.  When  a  train  is  approaching  a  station  at  a  velocity  of  40  ft. 
per  second,  a  mail  bag  is  thrown  at  right  angles  from  the  car  with 
a  speed  of  20  ft.  per  second.    Find  the  actual  direction  and  speed 
of  the  moving  bag. 

(Taken  from  Stone-Millis,  Plane  Geometry.) 

225.  The  resultant  of  three  forces  may  be  found  by  getting  the 
resultant  of  two  of  the  original  forces  and  then  finding  the  resultant 
of  that  and  the  third  original  force.    This  process  may  be  continued 
to  obtain  the  resultant  of  several  forces. 

Find  the  resultant  of  three  coplanar  forces  of  200  Ibs.,  150  Ibe., 
and  175  Ibs.  respectively,  acting  on  the  same  body  at  the  same 


72  PLANE  GEOMETRY 

time.  The  angle  between  the  first  and  second  force  is  45°,  and  that 
between  the  second  and  third  is  60°.  Why  note  that  the  forces 
are  coplanar? 

226.  A  force  of  100  Ibs.  makes  an  angle  of  60°  with  a  second 
force  of  120  Ibs.  exerted  on  the  same  body,  and  makes  an  angle  of 
90°  with  a  third  force  of  140  Ibs.,  and  an  angle  of  120°  with  a  fourth 
force  of  160  Ibs.  If  the  forces  are  coplanar  and  act  simultaneously, 
find  their  resultant. 

Theorem  20.  The  diagonals  of  a  parallelogram  bisect  each 
other. 

EXERCISES.    SET  XXVII  (continued) 

227.  Draw  any  line  through 
the  intersection  of  the  diagonals 
of  a  parallelogram. 

(a)  Give  a  list  of  the  pairs  of 
congruent    triangles   formed. 
Give    reasons  for   your    asser- 
tions. 

(b)  Give  a  list  of  the  pah's  of 
congruent  quadrilaterals.    Verify  your  statements. 

228.  Cut  a  parallelogram  out  of  cardboard.  Placing  a  pin  at  the 
intersection  of  the  diagonals,  try  to  balance  the  parallelogram. 
Why  would  you  expect  it  to  balance? 

The  intersection  of  the  diagonals  is  called  the  center  of  gravity 
of  a  parallelogram.  Why? 

Theorem  21.  A  quadrilateral  whose  diagonals  bisect  each  other 
is  a  parallelogram. 

EXERCISE.     Set  XXVII    (continued) 

229.  The  same  principle  is  often 
used  in  the  construction  of  iron 
gates  that  was  employed  in  the 
making  of  a  parallel  ruler  used  in 
the   eighteenth  century   (see   dia- 
gram).   What  is  the  principle? 

The  student  is  encouraged  to  make  such  an  instrument. 


,   POLYGONS  AND  PARALLELOGRAMS      73 

EXERCISES.    SET  XXVIII.    PARALLELS 

230.  Classify  quadrilaterals. 

231.  Summarize  ways  of  proving: 
(a)  Sects  equal. 

(6)  Angles  equal. 
(c)  Lines  parallel. 

LIST  OF  WORDS  DEFINED  IN  CHAPTER  m 

Exterior  angle  and  angles  of  a  polygon.  Convex  polygon,  diagonal;  quadri- 
lateral, pentagon,  hexagon,  heptagon,  octagon,  nonagon,  decagon;  regular 
polygon.  Parallelogram,  rectangle.  Distance  between  parallels. 

SUMMARY  OF  THEOREMS  DEVELOPED  IN  CHAPTER  in 

13.  The  sum  of  the  angles  of  a  triangle  is  a  straight  angle. 

Cor.  1.    A  triangle  can  have  but  one  right  or  one  obtuse  angle. 

Cor.  2.  Triangles  having  two  angles  mutually  equal  are  mutually 
equiangular. 

Cor.  3.  A  triangle  is  determined  by  a  side  and  any  two  homolo- 
gous angles . 

Cor.  4.  An  exterior  angle  of  a  triangle  is  equal  to  the  sum  of  the 
non- adjacent  interior  angles. 

14.  The  sum  of  the  angles  of  a  polygon  is  equal  to  a  straight  angle  taken 
as  many  times  less  two  as  the  polygon  has  sides. 

Cor.  1.  Each  angle  of  a  regular  polygon  of  n  sides  equals  the  ^2th 
part  of  a  straight  angle. 

Cor.  2.  The  sum  of  the  exterior  angles  of  a  polygon,  made  by  produc- 
ing each  of  its  sides  in  succession,  is  two  straight  angles. 

Cor.  3.  Each  exterior  angle  of  a  regular  polygon  is  the  |th  part 
of  a  straight  angle. 

15.  If  two  angles  of  a  triangle  are  equal,  the  sides  opposite  them  are  equal. 

Cor  1.    Equiangular  triangles  are  equilateral. 

16.  Either  diagonal  of  a  parallelogram  bisects  it. 

Cor.  1.    The  parallel  sides  of  a  parallelogram  are  equal,  and  the 

opposite  angles  are  equal. 
Cor  2.    Parallels  are  everywhere  equidistant. 

17.  A  quadrilateral  whose  opposite  sides  are  equal  is  a  parallelogram. 

18.  A  quadrilateral  having  a  pair  of  sides  both  equal  and  parallel  is  a 
parallelogram. 

19.  A  parallelogram  is  determined  by  two  adjacent  sides  and  an  angle. 

20.  The  diagonals  of  a  parallelogram  bisect  each  other. 

21.  A  quadrilateral  whose  diagonals  bisect  each  other  is  a  parallelogram. 


CHAPTER  IV 

AREAS 

A.  INTRODUCTION.    REVIEW  OF  FRACTIONS  * 

In  dealing  with  areas,  we  are  largely  concerned  with  ratios.  A 
ratio  is  a  fraction,  and  therefore  our  work  in  this  chapter  presup- 
poses a  familiarity  with  fractions.  For  those  of  us  who  need  a 
review  of  this  topic  the  following  will  be  helpful. 

A  fraction  is  an  indicated  quotient,  the  dividend  oi  which  is  the 
numerator  and  the  divisor  the  denominator. 

Since  this  is  review,  let  us  summarize  without  discussion  the 
fundamental  facts  which  we  need  to  recall  about  fractions. 

PRINCIPLES 

I.  The  value  of  a  fraction  is  not  changed  if  both  numerator  and 
denominator  (i.e.,  both  terms  of  the  fraction)  are  multiplied  or 
divided  by  the  same  quantity.  Why? 

This  statement  includes  cancellation,  for  that  is  the  process  of 
dividing  both  numerator  and  denominator  by  a  common  factor. 

Illustrations:        1.  £.  =  <*?  =  !^?  ^.    64o36  _  a2 

F  =  56  =  ^ 

3.        a2-4  o+2 


2a2-8a+8      2(a-2) 

II.  Considering  as  the  signs  of  a  fraction  that  of  the  numerator, 
that  of  the  fraction,  and  that  of  the  denominator,  any  two  may  be 
changed  without  changing  the  value  of  the  fraction.  Why? 

Recall  that  the  numerator  and  denominator  of  a  fraction  are 
treated  as  if  each  were  enclosed  in  a  parenthesis. 

Illustrations:    1.    a_  _       —a  _       _£_  -  -  .Zf?  2.   a— 2_  2— a_ 

b~         b    '         —b~—b  a—5~ 5—a~ 

3.   abc  _  a(—  6)c  _  —  abc  _ 

def  =  det^f)  =  ^def  = 

*  Those  who  have  not  studied  algebraic  fractions  are  advised  to  take  up  this 
topic  at  this  time.    Any  standard  algebra  text  will  furnish  sufficient  material. 
74 


AREAS  75 

III.  Fractions  may  be  added  or  subtracted  by  changing  them  to 
equivalent  fractions  having  the  same  denominators  and  adding  or 
subtracting  their  numerators,  putting  the  sum  or  difference  over  the 
common  denominator.  Why? 

Changing  to  a  same  denominator  depends  for  its  validity  upon 
preceding  principle  I.    The  same  or  common  denominator  may 
be   found    by  getting  the    Lowest    Common    Multiple  of  the 
denominators. 
Illustration  :        m—  2n          mz—  3n2     3m  —  n 


m3—n3       m—n 
The  L.  C.  M.  of  ra2+mrc+n2,  m3—  n3,  and  m—n  is  m3—  n3. 

_(m-2n)(m-n)      (m2-3n2)  .   (3ra-n)(m2+mn+n2) 

.  .  IIIG  Sliin  =  -  -  -  -  -  —    -  -  -  —    T  -  ;  -  -  -  • 


m3—  n3  m3—  n3 


m3-n3 


IV.  Fractions  may  be  multiplied  by  multiplying  their  numerators 
and  denominators  separately,  obtaining  thus  the  numerator  and  de- 
nominator respectively  of  the  product.    Why? 
Illustration:    5c2-20d2   c2-2cd+4dz 


5(c+2d)(c-2d) 

~  (c  +2d)  (c2-2cd+4d2)  '  ~      25cd* 
_c-2d  5 

"  5cd4 

V.  Fractions  may  be  divided  by  inverting  the  divisor  and  proceeding 
as  in  multiplication.    Why? 

Illustration:    a2-7a+12  ^a2-16 
o-l        ''    1-a2 

-1 
_  (o-4)  (o-3)    (l-o 


a-1       "•(a-4)(a+4)  4+a 

EXERCISES.    SET  XXIX.     FRACTIONS 

232.  Would  the  value  of  a  fraction  be  changed  if  both  numerator 
and  denominator  were  squared?    Illustrate  and  give  a  reason  for 
your  answer. 

233.  Why  in  dividing  fractions  do  we  invert  the  divisor  and 
multiply? 


76  PLANE  GEOMETRY 

234.  Would  the  result  of  cancellation  ever  be  zero?    Illustrate 
and  give  a  reason  for  your  answer. 

235.  Write  in  four  ways  the  fraction  ^g 

236.  Find  the  difference  between 

(a)  1   .    1  1          (b)  I       1  1  (c)n+l      An  , 

v'  —  h-and-  v/  ---  and-  '  -5-  and  --  +  1. 

n      m        n-\-m  n     m        n-m  3  6 

237.  Are  any  or  all  of  the  expressions  in  the  following  groups 
identical? 

(a)bh  L,  6,         ,  ,  h          (b)b+B  b  .B        ,1,,,  ™ 
~2  >  2  bh)  2  '  2  ~2~'  2  +  ~2'         2  (  +    ^ 

Justify  your  answers. 

238.  State,   with  reasons,   whether  or  not  the  following  are 
identities: 

(a)       1  1  (6)       1  1 

(5_a)2-(a_6)2  (6  -a)3      (a-6)3 


239.  Add:    3a  —  6          2         Why  is  it  correct  to  refer  to  such  a 

a2  —b2      b  —  a      combination  as  addition? 
Simplify   (which  means  do  whatever  is  indicated  by  the  symbols)  . 


241  3___ 

-  6xy    V      2-x 

-• 


4a2-4a6-362     /       96 2 


244. 
245. 
246. 


A+p      1+p' 

<  v2 


AREAS  77 


248.  (0)  *+-=  (6) 

a     a+b 


,  N    ,  [x+i  z-ii 

(c)  x+\ r-jLl 

I   a     a+bl 

249.  a-b-^^™ 


a-b 
250. 


a2ra2-a2ra-56a2 

Qr2-49)(:r2-16;r+63) 
' 


252 


m+2  ra  -3/  \4  -mV    m2+m  - 

B.  AREAS.    DEVELOPMENT  OF  FORMULAS 


Just  as  a  sect  is  measured  by  finding  the  number  of  linear  units 
it  contains,  so  a  surface  is  measured  by  finding  the  number  of 
square  units  it  contains,  or  better,  itsratio  to  the  unit  square,  which 
is  known  as  the  area  of  the  surface.  A  square  unit  is  a  square  each 
side  of  which  is  a  linear  unit.  For  example,  if  we  were  to  measure  the 
length  and  the  width  of  this  page  (taking  the  inch  as  a  linear  unit) 
a  square  inch  would  be  the  corresponding  square  unit,  and  the  area 
would  be  found  in  square  inches.  The  selection  of  the  unit  in 
practical  measurements  is  just  a  matter  of  convenience.  Why,  for 
instance,  select  an  inch  instead  of  a  mile  in  measuring  the  length 
of  this  page? 

In  comparing  areas  or  sects,  we  compare  the  abstract  numbers 
which  express  the  ratio  of  their  measures  in  terms  of  a  common 
unit.  When  we  say  that  two  sects  compare  as  5  to  4  we  refer  to  the 
fact  that  when  measured  in  terms  of  the  same  unit  their  measures 
would  compare  as  5  to  4  —  say  one  was  5"  and  the  other  4".  Sim- 
ilarly, when  we  say  two  rectangles  compare  as  6  to  5  we  mean  that  if 
the  area  of  one  were  6  sq.  ft.  the  area  of  the  other  would  be  5  sq.  ft. 

Such  quantities  as  we  have  just  referred  to  are  said  to  be  com- 
mensurable because  they  can  be  measured  in  terms  of  the  same  unit. 
In  comparing  two  sects  it  is  sometimes  impossible  to  get  a  common 
unit  of  measure,  that  is,  to  select  a  unit  so  that  it  will  be  contained 
an  exact  number  of  times  in  both.  Such  sects  are  said  to  be  incom- 


78  ,  PLANE  GEOMETRY 

mensurable.  It  is  known  from  experimental  work  in  mensuration 
that  a  circumference  and  its  diameter  are  incommensurable,  for  if 
the  diameter  were  1  inch,  the  circumference  would  be  3.  141  59  4- 
inches  (TT  inches). 

EXERCISES.    SET  XXX.    COMPARISON  OF  SECTS 

253.  Assuming  the  fact  that  the  square  of  the  hypotenuse  of  a 
right  triangle  is  equal  to  the  sum  of  the  squares  of  the  other  two 
sides,  show  that  the  diagonal  of  a  square  and  a  side  of  the  square 
are  incommensurable. 

254.  When  two  sects  are  commensurable  a  common  measure 
can  be  found  as  follows: 

^  _  X    zv    Zz    B      K  *h.e  sects  are  4^ 

'   andPQ.   Layoff  PQ 
on  AB.     It  will  be 


p  y  z     Q 

contained  once  with 

a  remainder  XB.  _Lay  off  XB_on  PQ.  It  will  be  contained  twice 
with  a  remainder  ZQ.  Lay  off  ZQ  onXB.  It  will  be  contained  just 
3  times  with  no  remainder.  Then  Z2B,  is  a  common  measure  (in- 
deed the  greatest  common  measure)  of  APand  PQ.  Any  part  of 
Z2B  would  also  be  a  common  measure  of  AB  and  PQ.  Call  Z2B,  u. 

How  many  u's  does  XB  contain? 

How  many  u's  does  YZ  contain? 

How  many  u's  does  AX  contain? 

Show  that  u  is  a  common  measure  of  AB  and  PQ. 

We  shall  refer  to  the  consecutive  sides  of  a  rectangle  as  its  dimen- 
sions, calling  one  the  altitude,  and  the  other  the  base. 

For  the  sak^  of  brevity  the  expressions  "the  ratio  of  two  sects" 
or  "the  products  of  two  sects"  will  be  used  to  indicate  the  ratio 
or  the  product  of  the  abstract  numbers  expressing  their  lengths 
in  terms  of  the  samo  unit  of  measure.  For  the  same  reason  "rec- 
tangle," "parallelogram,"  etc.,  will  be  used  for  the  "area  of  rec- 
tangle," "area  of  parallelogram,"  etc.  Therefore,  the  expression 
"two  rectangles  compare  as  the  products  of  their  dimensions"  is  a 
conventionally  abbreviated  form  of  "the  areas  of  two  rectangles 
compare  as  the  products  of  the  lengths  of  their  dimensions  ex- 
pressed in  terms  of  the  same  linear  unit." 


AREAS 


79 


Theorem  22.*    Rectangles  having  a  dimension  of  one  equal  to 
that  of  another  compare  as  their  remaining  dimensions 
A     E  D  PL ^T 


\\  * 

u\                   ,       , 

B 

F    G                       C 

M   N  S 

Given:  Rectangles  ABCD  (call  it  R)  and  PQST  (call  it  ft)  with  AB=PQ, 

and  BC  and  QS  commensurable. 
To  prove:  R_  =  BC 
Ri~  QS 
Suggestions  for  proof: 

Let  u  be  a  common  measure  of  £C  and  QS. 

Lay  uoff  on  "BC  and  Q£  and  erect  JLs  at  points  of  division  F,  G.  .  . 
M,N,....  — 

If  w  is  contained  m  times  in  BC  and  n  times  in  QS  what  is  the  ratio  —  ? 

What  kind  of  figures  are  ABFE,  .  .  .  ,  P£ML,  .  .  .  ?  QS  ' 

Are  they  congruent?   Why? 

If  R  and  ft  are  respectively  composed  of  m  and  n  congruent  parallelograms 

•n 

what  is  the  ratio  of  —  ? 
ft 

Theorem  23.    Any  two  rectangles  compare  as  the  products  of 
their  dimensions. 


Given:  Rectangles  R  and  ft  with  dimensions  h,  b,  and  hi,  bi  respectively. 
To  prove  :-~-  =  r-r- 

Ki        Oi/ll 

Suggestions  for  proof: 

Construct  a  rectangle  (ft)  having  as  dimensions  h  and  61. 

S^? ft^? -''ft^7 

*Although  the  proof  here  suggested  applies  only  to  those  cases  where  the 
sects  are  commensurable,  the  theorem  is  always  valid. 


80  PLANE  GEOMETRY 

Theorem  24.    The  area  of  a  rectangle  is  equal  to  the  product 

of  its  base  and  altitude. 
Given  :  Rectangle  R  with 

base  b  and  altitude  h. 
To  prove:   Area  of  R  =  bh. 
Suggestions  for  proof: 


b  What  does  area  mean? 

Call  the  unit  of  surface  u.    u  will  be  a  rectangle  with  base  and  altitude  1. 
^.     why?      For  what  does       stand? 


Why? 


EXERCISES.     SET  XXXI.    AREAS  OF  RECTANGLES 

255.  What  per  cent  of  surface  is  allowed  for  joints  and  waste, 
if  120  rectangular  sheets  of  tin,  14"  by  18",  are  just  sufficient  to 
cover  a  roof  165  sq.  ft.? 

256.  A  map  is  drawn  to  a  scale  of  1"  to  1000  miles.  What  actual 
area  would  be  represented  on  the  map  by  a  foot  square? 

257.  From  a  rectangular  sheet  of  paper  cut  a  strip  J^  of  the  sheet 
in  width.   What  part  of  the  sheet  is  left?   Then  from  the  same  sheet 
cut  off  J^  of  its  length.   What  part  of  the  original  sheet  is  now  left? 

258.  Prove,  by  letting  a  and  b  represent  the  lengths  of  two 
sects,  that  (a=*=6)2=a2+62±2a6. 

259.  How  long  would  a  rectangular  strip  of  paper  1  sq.  ft.  in 
area  be  if  it  were  .01"  wide? 

260.  Both  a  square  and  a  rectangle  three  times  as  long  as  it  is 
wide  have  a  perimeter  of  64  ft.    Compare  their  areas. 

261.  Thucydides  (430  B.C.),  a  Greek  historian,  estimated  the 
size  of  the  Island  of  Sicily  by  the  time  it  took  him  to  sail  around  it, 
knowing  how  long  it  took  him  to  sail  around  a  known  area.    Was 
his  method  correct?    Give  a  reason  for  your  answer. 

262.  144  sq.  ft.  is  the  area  of  a  square  and  also  of  a  rectangle 
four  times  as  long  as  wide.    How  do  their  perimeters  compare? 

//  one  side  of  a  parallelogram  is  selected  as  its  base,  the  distance 
between  it  and  the  opposite  side  is  called  its  altitude. 

Would  it  make  any  difference  at  what  point  its  altitude  was 
measured?  Why? 

How  many  altitudes  has  a  parallelogram? 

Is  this  definition  consistent  with  what  we  have  referred  to  as 
an  altitude  in  the  case  of  the  rectangle? 


AREAS 


81 


H 


Theorem  25.    The  area  of  a  parallelogram  is  equal  to   the 
product  of  its  base  and  altitude.  A  b  D 

Given :  EJABCD  with  base  6  and  altitude  A. 
To  prove:  Area  of  ElABCD^bh. 
Suggestions  for  proof: 

Draw  AH  and  DH i _L BC.    What  kind 
of  figure  is  AHHiD? 

What  is  the  area  of  AHHJ). 

Compare  A  ABH  and  DCHi. 

How  do  the  areas  of  AHHiD  and  ABCD  compare? 

What  is  the  base  of  each,  and  what  the  altitude  of  each? 

Suppose  AH  cut  BC  produced,  will  the  theorem  still  hold  true? 

EXERCISE.    SET  XXXII.    AREAS  OF  PARALLELOGRAMS 

263.  If  one  side  of  your  parallel  ruler  is  held  fixed  while  the 
opposite  side  is  raised  and  lowered  to  various  positions,  will  the 
areas  of  the  various  parallelograms  be  changing?    If  so,  what  will 
be  the  greatest  area  obtainable,  and  what  the  least? 

Any  side  oj  a  triangle  may  be  taken  as  its  base,  and  the  perpen- 
dicular from  the  opposite  vertex  to  that  side  will  be  its  altitude. 
EXERCISE.    SET  XXXIII.     ALTITUDES  OF  TRIANGLES 

264.  (a)  How  many  altitudes  has  a  triangle?    Illustrate. 

(6)  Does  your  answer  to  (a)  hold  for  a  right  triangle?    Illustrate. 

(c)  Will  the  altitudes  of  a  triangle  always  fall  within  the  triangle? 

(d)  What  fact  have  we  already  proved  about  the  altitudes  of  an 
equilateral  triangle? 

Theorem  26.    The  area  of  a  triangle  is  equal  to  half  the  product 

of  its  base  and  altitude. 
Given:  AABC  with  b  as  base  and 


_P, 


7 


h  as  altitude. 

To  prove  :   Area  of  AABC  = 
Suggestions  for  proof: 

DT&wBX  ||  CA  and  IT 
meeting  at  P. 


CB, 


What  kind  of  figure  is  APBCt 

What  is  the  area  of  APBC1 

How  is  the  area  of  AABC  related  to  that  of  APBC? 

Cor.  1.  Any  two  triangles  compare  as  the  products  of  their  bases 

and  altitudes. 
Cor.  2.  Triangles  having  one  dimension  equal  compare  as  their 

remaining  dimensions. 
6 


82 


PLANE  GEOMETRY 


EXERCISES.     SET  XXXIV.    AREAS  OF  TRIANGLES 
265.  Prove  theorem  26,  using  the  suggestion  given  by  the  accom- 
panying diagram.  S*^"\     #      ~7 

266.  Calculate  the  area 
of  the  letter  Z  shown  in 
the  figure,  the  dimensions 


being  indicated  in  centimeters. 

267.  With  a  marked  edge  draw  a  triangle,  and  tak- 
ing necessary  measurements  find  its  area,  using  in  turn  its  three 
bases  and  altitudes. 

268.  Where  do  the  vertices  of  all  triangles  having  the  same  base 
and  the  same  area  lie?  Give  reasons  for  your  answer. 

A  trapezoid  is  a  quadrilateral  with  one,  and  only  one,  pair  of  par- 
allel sides.  The  parallel  sides  are  called  its  bases,  and  the  distance 
between  them  is  called  its  altitude. 

Why  do  the  words  "and  only  one"  need  to  be  included  in  the 
definition? 

Theorem  27.  The  area  of  a  trapezoid  is  equal  to  half  the  product 
of  its  altitude  and  the  sum  of  its  bases. 

Given:   Trapezoid  ABCD  with  bases  6  and  bi 

and  altitude  h. 

To  prove:  Area  of  trapezoid  equals  \h  (6+61). 
Notes  on  proof: 

This  fact  may  be   proved  by  making  con- 
structions that  will  divide  the  figure  into 

rectangles,  parallelograms,  or  triangles,  or  combinations  of  them.  Why? 
The  following  diagrams  show  some  such  constructions.  The  student  may 
use  one  of  them  or  suggest  another.  It  is 
interesting  as  an  exercise  to  show  that  the 
same  formula  for  the  area  of  a  trapezoid 
may  be  derived  from  each  of  the  following 

diagrams.     Which  gives  the  simplest  deri-     if  C 

vation?  1 


0 


2.  DrawAtf  \\XY. 


B  D  Y 

3.  Draw  ZfiJLXF  and  ~CD LXY 


AREAS 


4.  Draw  PQ  and  MN  ±AB 
produced. 

B 


5.  Through  N,the  midpoint  of  #(7,  draw 
WN  ||  DA  cutting  ~AB  produced  in  P. 
B 


_      _  _^  ~0 

6.  Extend  A£by  DC  to  E  and  DC     7.  Draw  CX  \\  DA  cutting  AB  produced 

by  AB  to  F.  in  F. 

8.  Extend  ~YX  to  A_  p  ^ 

R  so  that  XR=AB. 

Draw_RZ  ||  FA.  Bi-  ,.,./  \P 

sect  AF  in  M  and 

draw  WP  1 1  FX"  cut- 

ting  1Z  in  Q.  Y  X~  R 

EXERCISES.    SET  XXXV.    AREAS  OF  TRAPEZOIDS 

269.  The  diagram  shows  how  the  area  of  an  irregular  polygon 
may  be  found,  if  the  distance  of  each  vertex 
from  a  given  base  line,  as  XY,  is  known. 
These  distances  AAi,  BBi,  etc.,  are  called 
offsets,  and  are  the  bases  of  trapezoids 
whose  altitudes  are  A\B\t  Bid,  C\D\,  etc. 
The  area  ABCDEF  may  now  be  found 
by  the  proper  additions  and  subtractions. 
On  cross-section  paper  plot  the  points 
whose  coordinates  are  given  below,  join 
them  in  order,  draw/'7  A,  and  find  the  inclosed  area  in  each  case: 


X 


Case 

A 

B 

C 

D 

# 

F 

(a) 

5,7 

4,5 

4,1 

7,0 

5,3 

6,3 

(b) 

2,7 

3,3 

6,0 

5,2 

6,6 

8,7 

(c) 

2,5 

3,6 

2,4 

4,  2           6,  5 

4,7 

84 


PLANE  GEOMETRY 


d    d    d    d    d    d     d 


270.  In  order  to  determine  the  flow  of  water  in  a  certain  stream, 
soundings  are  taken  every  6  ft.  on  a  line  AB  at  right  angles  to  the 
current.  A  diagram  may  then  be  made  to  represent  a  vertical 
cross-section  of  the  stream.  If  the  area  of  this  cross-section  and 

the  speed  of  the  current  are  known, 
it  is  possible  to  determine  the 
amount  of  water  flowing  through 
the  cross-section  in  a  given  time. 
The  required  area  is  often  found 
approximately  by  joining  the  ex- 
tremities of  the  offsets  y0,  yi}  y2,  etc.,  by  straight  lines,  and  finding 
the  sum  of  the  trapezoids  thus  formed.  That  is,  the  strips  between 
successive  offsets  are  replaced  by  trapezoids.  This  gives  the  Trape- 
zoidal Rule  for  finding  an  area.  It  may  be  stated  as  follows:  To 
half  the  sum  of  the  first  and  last  offsets  add  the  sum  of  all  inter- 
mediate offsets,  and  multiply  this  result  by  the  common  distance 
between  the  offsets. 

(a)  State  this  as  an  algebraic  formula. 
(6)  Verify  the  formula. 

(c)  Find  the  area  of  the  cross-section  of  a  stream  if  the  soundings 
taken  at  inter\7als  of  6  ft.  are  respectively  5  ft.,  6.5  ft.,  11  ft.,  14.5 
ft.,  16  ft.,  9  ft.,  and  6.5  ft. 

(d)  In  the  midship  section  of  a  vessel  the  widths  taken  at 
intervals  of  1  ft.  are  successively  16,  16.2,  16.3,  16.4,  16.5,  16.7, 
16.8,   15,  10,  4,  0,  measurements  being  in  feet.     Find  the  area 
of  the  section.     (Use  the  line  drawn  from  the  keel  ±  to  the  deck 
as  base  line.) 

6271.  A  third  rule  for  finding  plane  areas,  known  as  Simpson's 
Rule,  usually  gives  a  closer  result  than  the  Trapezoidal  Rule.  In 
proving  Simpson's  rule  two  consecutive 
strips  are  replaced  by  a  rectangle  and 
two  trapezoids  as  follows:  Divide  2d 
into  three  equal  parts,  erect  J_s  at  the 
points  of  division,  and  complete  the  rec- 
tangle whose  altitude  is  the  middle 
offset  2/1,  as  in  the  figure.  Join  the 
extremities  of  y  and  y2  to  the  nearer  upper  vertex  of  this  rectangle. 


AREAS  85 

Then  if  the  areas  of  the  trapezoids  are  T  and  Ti,  and  if  the  area 
of  the  rectangle  is  R, 


If,  now,  the  number  of  strips  is  even,  and  if  the  offsets  are  lettered 
consecutively  y0,  y\,  y%,  .  .  .  .,  yn,  the  addition  of  the  areas  of  suc- 
cessive double  strips,  found  by  the  above  formula,  gives  the  result  : 


In  words  :  To  the  sum  of  the  first  and  last  offsets  add  twice  the 
sum  of  all  the  other  even  offsets  and  four  times  the  sum  of  all  the 
odd  offsets,  and  multiply  by  one-third  the  common  distance 
between  the  offsets. 

(a)  Verify  this  rule. 

(6)  By  Simpson's  Rule  find  the  area  of  the  stream  in  Ex.  270  (c)  . 

(c)  By  Simpson's  Rule  find  the  area  of  the  section  of  the  vessel 
in  Ex.  270  (d). 

LIST  OF  WORDS  DEFINED  IN  CHAPTER  IV 

Ratio,  fraction,  numerator,  denominator,  terms  of  fraction,  cancellation, 
simplify.  Area,  commensurable,  incommensurable;  dimensions,  base  and 
altitude  of  rectangle,  parallelogram,  triangle,  trapezoid. 

SUMMARY  OF  THEOREMS  PROVED  IN  CHAPTER  IV 

22.  Rectangles  having  a  dimension  of  one  equal  to  that  of  another  compare 
as  their  remaining  dimensions. 

23.  Any  two  rectangles  compare  as  the  products  of  their  dimensions. 

24.  The  area  of  a  rectangle  is  equal  to  the  product  of  its  base  and  altitude. 

25.  The  area  of  a  parallelogram  is  equal  to  the  product  of  its  base  and 
altitude. 

26.  The  area  of  a  triangle  is  equal  to  half  the  product  of  its  base  and  altitude. 

Cor.  1.  Any  two  triangles  compare  as  the  products  of  their  bases 

and  altitudes. 
Cor.  2.    Triangles  having  one  dimension  equal  compare  as  the  re- 

maining dimensions. 

27.  The  area  of  a  trapezoid  is  equal  to  hah"  the  product  of  its  altitude  and 
the  sum  of  its  bases. 


CHAPTER  V 

ALGEBRA  AS  AN  INSTRUMENT  FOR  USE  IN  APPLIED 
MATHEMATICS 

A.  LOGARITHMS 

Although  logarithms  are  introduced  at  this  point,  as  a  part  of 
a  chapter  on  Algebra,  it  is  not  essential  that  they  be  studied  at 
this  time.  They  might  well  be  omitted  until  there  is  a  feeling  of 
necessity  on  the  part  of  the  pupils  in  the  solution  of  the  more  diffi- 
cult problems  based  upon  similarity  and  the  trigonometric  func- 
tions. In  fact,  those  schools  wishing  to  omit  such  problems  under 
Ratio,  Proportion,  Variation  and  Similarity  need  not  include  the 
topic  at  all.  For  this  reason,  when  logarithms  are  desirable  in 
the  solution  of  problems,  this  fact  has  been  indicated.  The  integ- 
rity of  the  course  will  not  be  injured  in  the  least  by  the  omission 
of  the  topic  of  logarithms  or  any  of  these  exercises. 

I.  INTRODUCTION 

If  82  =  64,  then  2  is  called  the  logarithm  of  64  to  the  base  8. 
This  is  written  Iog864  =  2.  Again  34  =  81  or  Iog381  =  4.  Hence  we  see 
that  the  logarithm  of  a  number  is  simply  the  exponent  of  the  power 
to  which  another  number  (called  the  base)  must  be  raised  to  obtain 
that  number.  Thus,  in  the  last  example  4  is  the  exponent  of  the 
power  to  which  the  base  3  must  be  raised  to  obtain  the  number  81. 

EXERCISES.     SET  XXXVI.     MEANING  OF  LOGARITHMS 

272.  Write  in  logarithmic  form: 

(a)  53  =  125    (c)     72  =  49    (e)  101-30103  =  20    (g)  103.47712  =  3000 
(6)  27=128  (d)  10*  =  10   (/)   103=1000       (h)  105-4?71  =  300,000 
(i)  106  =  1,000,000  (j)  lOO^lOOO 

273.  Write  in  exponential  form: 

(a)  Iog6216  =  3  (d)  Iog102-.3010  (g)  logw.01   =  -2 

(6)  Iog981=2  (e)  loglol  =  0  (h)  Iogi0.001=  -3 

(c)   Iognl331  =  3  (/)  Iogi0.l=-l  (i)   Iog10500  =  2.6990 
86 


ALGEBRA  IN  APPLIED  MATHEMATICS  87 

Before  studying  logarithms,  their  principles  and  applications,  it 
will  be  well  for  us  to  recall  the  laws  concerning  exponents  in  multi- 
plication, division,  involution  and  evolution.  We  have  learned  to 
multiply,  divide,  raise  to  powers  and  extract  roots  of  simple 
expressions  when  the  exponents  have  been  positive  integers,  so 
that  we  shall  here  only  summarize  what  we  already  know,  adding 
the  statement  that  the  laws  governing  positive  integral  exponents 
govern  all  exponents,  both  fractional  and  negative 

a.  PRINCIPLES  OF  EXPONENTS 

1.  The  exponent  of  the  product  of  any  number  of  factors  of  like 
base  is  equal  to  the  sum  of  the  exponents  of  the  factors. 

Illustrations:   (1)  an-ap-aq=an+p+q 

(2)  a*'a*=a* 

(3)  b-n'b~b=b-n-p 

(4)  b  +H  ~  A  =  6*  ~  A  =  6"  =  6* 

2.  The  exponent  of  the  quotient  of  two  quantities  of  like  base  is 
equal  to  the  exponent  of  the  dividend  diminished  by  that  of  the  divisor. 
Illustrations  :   (1)  x<*-x  —b  =xa  -b 

(2)  xa+ 


(3)  m     g-r-m3 

3.  The  exponent  of  any  power  of  a  quantity  is  equal  to  the  product 
of  the  exponent  of  the  base  and  the  index  of  the  power. 

Illustrations:   (1)  (6p)fl  =  6M 

/    p\r        pr 

(2)  \a*)°=a<s  ^ 

4.  The  exponent  of  any  root  of  a  quantity  is  equal  to  the  quotient 
of  the  exponent  of  the  base  and  the  index  of  the  root  when  that  index 
is  not  zero. 

Illustrations  :    (1)    _m       _       _* 
Vbx=b     m 

p  _aq 

(2)  «Vx~a=x     p 


88  PLANE  GEOMETRY 

EXERCISES.    SET  XXXVII.     DRILL  IN  APPLICATION  OF  LAWS 

OF  EXPONENTS 

Perform  the  operations  indicated  in  the  following  problems: 
274.  x*x  ~7x~8  282.  103-4362- 10-7856 

275.|ai|a*  283.10 

noA     in 

97ft      kn+2_-_Qk2  6<y±»    1U 

A I  U.    n/  •   On/ 

m     _n        J_  285.     (ID5'6723)7 

277.  pnpmp  nm  ,    *  \(x- 

(       _7,n  286.   (d*+»; 

278.  (rsatb)(-qs«    ~  b) 

287.  ^ 


279. 

(Hint:  Express 4  and  8  as 

powers  of  2  so  as  to  have  %n(3n  ~~  *  V 

a  common  base).  288. 

280.  25^125* 

289.* 

281.  25"^5~ 

*  NOTE  :  No  doubt  some  of  us  are  by  this  time  curious  to  know  the  mean- 
ing of  the  zero,  negative  and  fractional  exponent. 

(I)  Meaning  of  the  zero  exponent .  (II)  Meaning  of  the  negative  exponent . 

Thenx'ap=a°-ap  Then  a~p-ap=x-ap    Why? 

Euta~p-ap=l         Why? 
.:x-ap=l          Why? 


,.a-P^-         Why? 
Quote  the  axiom  applied.  o 

(III)  Meaning  of  the  fractional  exponent. 

The  fourth  law  given  in  the  text  expresses  the  meaning  of  the  fractional 
exponent  since  -$/xV=x?,  but  we  may  make  the  meaning  still  clearer  by  the 
following: 

Since  z*-:r*=.r,.r*  must  by  definition  of  square  root  be  the  square  root  (\/x\ 


ALGEBRA  IN  APPLIED  MATHEMATICS  89 

Let  us  see  how  the  annexed  table  may  be  used  to  simplify  cer- 
tain calculations. 

TABLE  OF  POWERS  OF  2 

2i  =  2  26  =  64  211  =  2048  216  =  65536 

22=4  27  =128  212=4096  217  =  131072 

23=8  28  =256  213=8192  2«  =  262144 

2^  =  16  29  =512  214  =  16384  219=  524288 

25  =  32  210  =  1024  215  =  32768 

1.  Suppose  we  wished  to  multiply  1024  by  512 
V  1024  =  210  and  512  =  29  and  210-29  =  219, 

.'.  1024-  512  =  219  =  524288. 

2.  Suppose  we  wished  to  divide  1,048,576  by  32768. 
V  1048576  =  220  and  32768  =  215  and  220 -j- 215  =  25, 
.'.  1048576  -r-  32768  =  25  =  32. 

3.  Suppose  we  wished  to  raise  64  to  the  third  power. 
V  64  =  26  and  (26)3  =  218  and  218  =  262144, 

/.  643  =  (26)3  =  262144. 

4.  Suppose  we  wished  to  find  the  5th  root  of  1,048,576. 
Y  1048576  =  22Q 

.'.  \/1048576  = 

In  similar  manner  a  table  of  the  powers  of  any  number  may  be 
computed,  and  these  four  operations  (multiplication,  division, 
involution,  evolution),  reduced  to  the  operations  of  addition,  sub- 
traction, multiplication,  and  division  of  exponents. 

i   i   i 

Similarly  xa-xs-x«- to  s  factors 

^1+1+J.-..  to  .terms.  ^ 

=xl=x  Why? 

.'.x~s  =  \^  Why? 

E      I  1  1 
Still  more  generally,  x9==x^-x^-x^ to  p  factors. 


But  v  xp  also  means  ^  )q  =  (x-x-x- to  p  factors)^. 

111 
=xQ-xQ-x9- to  p  factors. 


E 
=xi 


90 


PLANE  GEOMETRY 


EXERCISES.     SET  XXXVIII.     USE  OF  TABLES  OF  POWERS 
Using  the  given  table  of  2's,  find  the  values  of  the  following: 

290.  512X2048  292.  (32) 3  295.  A/ 262144 

256X16384          293.  (512): 


291. 


262144 
297 


294.  A/131072 


296.  A/65536 


.      4/(256)5(524288): 
*    V          (32)6 


298.  [(22)2]: 

299.  222' 


«    TABLE  OF  POWERS  OF  16 


No. 

Power 

No. 

Power 

1 
2 
4 
8 
16 
32 

0.00 
0.25 
0.50 
0.75 
1.00 
1.25 

64 
256 
1024 
4096 
65536,  etc. 

1.50 

2.00 
2.50 
3.00 
4.00 

303. 


65536(256)' 


EXERCISES.     SET  XXXVIII  (concluded) 
From  the  foregoing  table  compute  the  following: 

300.  16X4096 

301.  84 

302.  65536-5-4096  304-  16>< 

b.  HISTORICAL  NOTE 


64(1024) 
3)(65536X1024)2 


2562 


There  is  a  large  amount  of  computation  necessary  in  the  solution  of  some 
of  the  practical  applications  of  mathematics.  The  labor  of  making  extensive 
and  complicated  calculations  can  be  greatly  lessened  by  employing  a  table  of 
logarithms.  About  the  year  1614  a  Scotchman,  John  Napier  (1550-1617), 
Baron  of  Merchiston,  invented  a  system  by  which  multiplication  can  be  per- 
formed by  addition,  division  by  subtraction,  involution  by  a  single  multiplica- 
tion, and  evolution  by  a  single  division.  From  Henry  Briggs  (1556-1631), 
who  was  a  professor  at  Gresham  College,  London,  and  later  at  Oxford,  this 
invention  received  modifications  which  made  it  more  convenient  for  ordinary 
practical  purposes. 

Laplace,  the  great  French  astronomer,  said:  "The  employment  of  logar- 
ithms by  reducing  to  a  few  days  the  labors  of  months,  doubles,  as  it  were,  the 
life  of  an  astronomer,  besides  freeing  him  from  the  errors  and  disgust  insepar- 
able from  long  calculations." 


ALGEBRA  IN  APPLIED  MATHEMATICS  91 

The  logarithms  now  in  general  u^e  are  known  as  common  logarithms,  or  as 
Briggs'  logarithms,  in  order  to  distinguish  them  from  another  system,  also  a 
modified  form  of  Napier's  system.  The  logarithms  of  this  other  modified 
system  are  frequently  employed  in  higher  mathematics,  and  are  known  as 
natural  or  hyperbolic  logarithms. 

H.  PRINCIPLES  OF  COMMON  LOGARITHMS 

For  practical  purposes,  the  exponents  of  the  powers  to  which  10, 
the  base  of  our  decimal  system,  must  be  raised  to  produce  various 
numbers  are  put  in  table  form.  That  is,  the  logarithms  of  numbers 
to  the  base  10  are  tabulated.  For  the  positive  integral  powers  of 
10  we  would  need  no  tables,  for  those  we  can  find  by  inspection. 
But  exponents  may  be  negative  and  they  may  be  fractional.  For 
the  negative  integral  powers  of  10  as  we  shall  see  presently,  we 
would  need  no  tables  either.  But  fractional  exponents  or  the  frac- 
tional parts  of  exponents  we  cannot  readily  find,  and  hence  for 
them  we  need  tables. 

We  all  know  that  Similarly  it  can  be  shown  that 

Y103  =  1000,  .'.*log  1000=3.  V  10  -1  =  Jo>    .'•  log  .1     =  -1 

Y102=100,    /.  log  100   =2.  VlO-2=  —5,  .'.  log  .01    =  -2 

MO1  =  10,      /.  log  10     =  1.  YlO-3  =  ^-3,  .'.  log  .001=  -3 

101  101 

V         =10i-i=  10°,  and       =  1;  .'.  10"  =1,  /.  log  1  =  0. 


10-47712  or  iQTfrj  that  is,  the  one-hundred-thousandth  root 
of  10-47712  is  nearly  3.  .'.  log  3  =  .47712  nearly. 

Although  log  3  can  never  be  expressed  exactly  as  a  decimal 
fraction,  it  can  be  found  to  any  required  degree  of  accuracy.  In 
this  book  logarithms  are  given  to  four  decimal  places.  These  are 
sufficient  for  ordinary  computations. 

*  When  the  base  10  is  used  the  base  is  not  indicated  in  writing  the  loga- 
rithms of  numbers.  Thus  we  write  log  3  =  .47712,  not  log  103  =  .47712. 


92  PLANE  GEOMETRY 

EXERCISES.     SET  XXXIX.     COMMON  LOGARITHMS 

305.  What  are  the  logarithms  of  the  following  to  the  base  10: 

(a)  10000?  (c)  .0001?  (/)    \/10? 

(V       1     9  W)  109?  (g)  10*? 

'   10000*  (6)  10 -9?  (h)  10*? 

306.  Between  what  two  consecutive  integers  does  the  logarithm 
of  each  of  the  following  numbers  lie?    Why? 

(a)  600  (c)   13  (e)  46923 

(6)  5728  (d)  496,287  (/)  9 

307.  Between  what  two  consecutive  negative  integers  does  the 
logarithm  of  each  of  the  following  numbers  lie?    Why? 

(a)  .06  (c)  .0008  (e)  .00729  (g)  0.5 

(6)  .007  (d)  .0625  (/)  .00084 

308.  What  is  meant  by  saying  that: 

(a)  log  880  =  2.94448?  (6)  log  92.12  is  1.96435? 

(c)   log  4.37  is  .64048? 

Since  3585  lies  between  1000  and  10,000,  its  logarithm  lies 
between  3  and  4.  It  has  been  calculated  as  3.55449.  The  integral 
part  3  is  called  the  characteristic,  and  the  decimal  part  .55449,  the 
mantissa  of  the  logarithm. 

Y  358.5  =  3585  MO,  .'.  log  358.5  =  log  (3585  MO)  =  log  3585- 
log  10  =  3.55449  -1  =  2.55449. 

That  is,  since  log  3585  =  3.55449 

log  358.5  =  2.55449  and  similarly  it  can  be  shown 
that  log  35.85  =  1.55449 
log  3.585  =  0.55449 
log  .3585  =  .55449-1* 
log  .03585  =  .55449 -2. 
Thus  we  see  that 

(a)  The  characteristic  can  be  found  by  inspection  in  all  cases. 
'.'the  number  589  lies  between  100  and  1000,  log  589  lies  between 
2  and  3 .    . ' .  log  589  =  2 + some  mantissa. 

*Log  0.3585  =  .55449  —  1  may  be  written  in  two  other  ways  as  follows: 
1.55449  or  9.55449  —  10.  The  last  method  is  the  most  practical,  as  we  shall 
see  as  we  proceed. 


ALGEBRA  IN  APPLIED  MATHEMATICS  93 

(b)  The  mantissa  is  the  same  for  any  given  succession  of  digits, 
wherever  the  decimal  point  may  be. 

(See  last  table  of  numbers  with  their  logarithms.) 

(c)  Asa  result  of  (a)  and  (b)  only  a  table  of  mantissas  need  be  given. 

EXERCISES.    SET  XXXIX  (continued) 

309.  What  is  the  characteristic  of  the  logarithm  of: 

(a)  384?         (c)   .297?        (e)  A  number  of  n  integral  places? 
(6)  5286?        (d)  Any  number  of  millions? 
(/)   Any  decimal  fraction  whose  first  significant  digit  is  in  the 
first  decimal  place? 

(g)  In  the  second  decimal  place?      (h)  In  the  third  decimal  place? 
(i)  In  the  seventh  decimal  place?    (j)  In  the  nih  decimal  place? 

310.  From  the  last  exercise  formulate  a  principle  by  means  of 
which  the  characteristic  of  the  logarithm  of  any  positive  number 
may  be  found. 

HI.  THE  FUNDAMENTAL  THEOREMS  OF  LOGARITHMS 

(a)  The  logarithm  of  the  product  of  two  numbers  equals  the  sum  of 
their  logarithms  to  the  same  base. 

1.  Leta^lO*1,      then  log  a=h 

2.  Let  &=aoh,      then  log  b  =  k 

3.  .*.  ab=Wh+l2,  and  log  o&  =  Zi+k=log  a-f  log  b. 

(b)  The  logarithm  of  the  quotient  of  two  numbers  equals  the  logarithm 
of  the  dividend  minus  the  logarithm  of  the  divisor,  all  to  the  same  base. 

1.  Leta=10u  then  log  a=h 

2.  Let  6  =  10e,  then  log  b=l2 

3.  .'.     5"— Jofe=10Zl~S  and  log  -^-Z^log  a  -log  b. 

(c)  The  logarithm  of  the  nth  power  of  a  number  equals  n  times  the 
logarithm  of  that  number. 

1.  LetaH=10z,  then  log  a=l 

2.  .'.  an=10'n,  and  log  an=nl=n  log  a. 

(d)  The  logarithm  of  the  nth  root  of  a  number  equals  ^th  of  the 
logarithm  of  the  number. 

1.  Leta=10',  then  log  a=l 

i         i  i      i      i 

2:  ,",  a»^10n,  and  log  a*  =  n  =  ~n  log  a. 


94 


PLANE  GEOMETRY 


Th.  (c)  might  have  been  stated  more  generally,  so  as  to  include 

-     x 
Th.  (d)  thus:  Log  ay  =~  log  a.    The  proof  would  be  substantially 

the  same  as  in  Ths.  (c)  and  (d). 

EXERCISES.     SET  XXXIX  (concluded) 

Given  log  2  =  0.3010,  log  3  =  0.4771,  log  5  =  0.6990,  log  7  =  0.8451, 
and  log  514  =  2.7110,  find  the  following: 

311.  Log  6.         312.  Log  14.      313.  Log  710.      314.  Log  A/2- 

315.  Log  42.       316.  Log  5*.       317.  Log  105.    318.  Log  1.05. 

319.  Log  V514-  320.  Log  5142.  321.  Log  1542.  322.  Log  257^ 

323.  Logl799[=log(i-514-7)].     324.  Log\/3l    325.  Log\/21. 

326.  Show  how  to  find  log  5,  given  log  2. 
IV.  USE  OF  THE  TABLE 

(a)  Given  a  number,  to  find  its  logarithm. 

In  the  table  on  p.  103  only  the  mantissas  are  given.  For  in- 
stance, in  the  row  beginning  43,  and  in  columns  headed  0,  1,  2, 
3, ,9  will  be  found : 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

This  means  that  the  mantissa  of  log  430  is  .6335,  of  log  431  is 
.6345,  and  so  forth,  to  log  439. 

Therefore  log  431  =  2.6345,  log  434  =  2.6375,  log  43.7  =  1.6405, 
log  4.39  =  0.6425,  log  .438  =  .6415  -1,  log  .0433  =  .6365  -2.  Now, 
since  437.8  is  .8  of  the  way  from  437  to  438,  .'.log  437.8  is  about 
.8  of  the  way  from  log  437  to  log  438.  .'.  log  437.8  =  log  437+.S 
of  the  difference  between  log  438  and  log  437.  .'.  log  437.8  = 
2.6405+.8  of  .0010  =  2.6405+.0008  =  2.6413. 

This  process  of  finding  the  logarithm  of  a  number  lying  between 
two  tabulated  numbers  is  called  interpolation.  This  is  not  wholly 
accurate,  since  the  numbers  do  not  vary  as  their  logarithms,  but 
it  is  sufficiently  accurate  for  most  practical  purposes.  If  greater 
accuracy  is  desired,  tables  of  five  or  six  or  even  more  places  are 
used.  The  mantissas  here  given  are  correct  to  .0001.  This  will 
give  a  result  which  is  correct  to  three  figures  in  general,  and  an 
approximation  to  four  figures,  which  will  be  sufficiently  accurate 
for  the  computations  in  this  book. 


ALGEBRA  IN  APPLIED  MATHEMATICS 


95 


EXERCISES.     SET  XL.     USE  OF  TABLE 
Find  by  using  the  table: 


327.  Log  49.  332.  Log  14.7. 

328.  Log  723.  333.  Log  14.73. 

329.  Log  1580.          334.  Log  5.93. 

330.  Log  4285.          335.  Log  .00432. 

331.  Log  14.5.  336.  Log  1.672. 

(6)  Given  a  logarithm  to  find  the  corresponding  number. 
The  number  corresponding  to  a  given  logarithm  is  called  its  anti- 
logarithm.    Example:    Y  0.4771  =  log  3,  .*.  antilog  0.4771=3. 


337.  Log  .00002376. 

338.  Log  V29. 

339.  Log  5.6923. 

340.  Log  -v/36.54. 

341.  Log  .00576. 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

Here  we  see  that  antilog  4.6345  =  43100,  antilog  0.6395  =  4.36, 
antilog  3.6405  =  .00437. 

Now  suppose  we  wished  to  find  antilog  2.6417. 

V  2.6417  is  .2  the  way  from  2.6415  to  2.6425. 

.'.  antilog  2.6417  is  about  .2  the  way  from  antilog  2.6415  to  anti- 
log  2.6425. 

.'.  antilog  2.6417  is  about  .2  the  way  from  .0438  to  .0439. 

.*.  antilog  2.6417  =  .04382. 

The  following  form  is  a  good  one  for  use. 

Required:  Antilog  2.7361.  Antilog  2.7364  =  545  2.7361 

Antilog  2.7356  =  544  2.7356 

Tabular  diff .  = 


8 

.'.  Antilog  2.7361  =  5441  =  544.6. 


Diff. 


EXERCISES. 
Find  from  the  table: 

342.  Antilog  1.9321. 

343.  Antilog  2.9049. 

344.  Antilog  2.7813. 

345.  Antilog  3.0354. 

346.  Antilog  1.0354. 

347.  Antilog  3.1628. 

348.  Antilog  4.8393. 

349.  Antilog  10.5843. 


SET  XL.  (concluded) 


350.  Antilog  0.6923 -2. 

351.  Antilog  8.6923 -2. 

352.  Antilog  7.5194 -10. 

353.  Antilog  9.2490 -10. 

354.  Antilog  10.4687 -10. 

355.  Antilog  3.5357. 

356.  Antilog  0.3471. 


96  PLANE  GEOMETRY 

(c)  Computation  by  logarithms. 

Since  many  errors  occur  because  of  failure  to  arrange  work  care- 
fully, the  pupil  is  advised  to  arrange  all  work  in  as  compact  and 
neat  a  form  as  possible.  A  few  examples  worked  out  in  full  may 
be  suggestive,  therefore  we  append  the  following: 

1.  In  how  many  years  will  $600  double  itself  at  3%  interest 
compounded  annually? 

Solution:   Let  the  number  of  years  be  n. 

At  the  end  of  one  year  the  amount  will  be  1.03  of  $600,  at  the 
end  of  the  second  year  it  will  be  1.03  of  1.03  of  $600,  or  1.032 
of  $600,  and  so  forth. 

/.  at  the  end  of  n  years  it  will  be  1.03n  of  $600. 

/.  1.03* X600  =  1200  or  1.03"  =  2 

/.  n  log  1.03  =  log  2  and  .'.  ft^.j^gj 

log2  =  .3010J.      =.3Q10 
log  1.03  =  .0128j  .0128 

.'.  log  n=  log  .3010  -log  .0128 
log  .3010  =  9.4786 -10 
log  .0128  =  8. 1072 -10 

.'.  log  n=  1.3714  antilog  1.3714  =  23.5+ 

*.  n  =  23.5+,  or  the  sum  will  double  itself  in  24  years. 

0    ^       .  -    5286X\/427 

2.  Required  the  value  of  3  .        

\/1754X3292 

Solution : 
log  529  =  2.7235 
log  528  =  2.7226 
Tab.  diff.      =9 

A 

5.4.'.  log  528.6_  =  2.7231 
log  427  =  2.6304  .'.  log  \/427=  1.3152 

/.log  numerator  =  4.0383 14.0383  -10 

log  1760  =  3.2455 
log  1750  =  3.2430 
Tab.  diff.  =  25 

_^4 

10.0 

' 


ALGEBRA  IN  APPLIED  MATHEMATICS  97 


/.log  1754  =  3.2440  .'.  log  A/1754  =1.0813; 

log  numerator  =  14.0383-10 

log  3300  =  3.5185 
log  3290  =  3.5172 
Tab.  diff.=  13 

J2, 

2.6  or  3.  '.log  3292  =3.5175 

/.  log  denominator  =  4.5988  .........  4.5988 

antilog  9.4409  -  10  =  .276          .'.  log  fraction  (log 

numerator—  log  denominator)  =9.  4395  -  10 
antilog  9.4393-10  =  .275  9.4393  -  10 

Diff.  for  1  =  16  2 

/.  antilog  9.4395  -10  =  .275A=.275|. 
.'.  Fraction  =  .2751 

As  soon  as  the  pupil  is  able  to  interpolate  mentally  the  left 
column  may  be  omitted. 

Often  in  problems  involving  the  process  of  evolution  difficulties 
may  arise  owing  to  the  fact  that  the  characteristic  may  be  negative 
and  not  a  multiple  of  the  divisor,  while  the  mantissa  is  always 
positive.    For  instance,  it  may  be  desired  to  find  \/0.03. 
log  </003  =  i-  log  0.03 

=  -g-  (2.  477  1  )  .   Why  is  this  a  decidedly  impracticable 

form? 
=  |  (8.4771  -10).     Why  is  this  an  inconvenient 

form? 
'  =  i  (28.4771  -30).     Why  is  this  the  form  which  is 

adopted? 
=  9.4924-10 
.'.  ^0.03  =  0.3107 
Again,  suppose  \/0.34  is  called  for. 
log  0.3=1.4771 

.Mog\/6~3~4  =£(9.4771-  10)  =5L^L  *?.  Now,  40  is  not  a 
multiple  of  10  and  7,  hence  we  change  to  the  equivalent 


.'.  antilog  9.7012  -10= 

7 


98  PLANE  GEOMETRY 

In  place  of  a  table  of  logarithms  engineers  often  use  an  instru- 
ment called  a  "  slide  rule."  This  is  really  a  mechanical  table  of 
logarithms  arranged  ingeniously  for  rapid  practical  use.  Results 
can  be  obtained  with  such  an  instrument  far  more  quickly  than 
with  an  ordinary  table  of  logarithms,  and  that  without  recording 
or  even  thinking  of  a  single  logarithm.  A  "  slide  rule"  ten  inches 
long  gives  results  correct  to  three  figures.  In  work  requiring  greater 
accuracy  a  larger  and  more  elaborate  instrument  which  gives  a 
five-figure  accuracy  is  used.* 

EXERCISES.     SET  XLI.     COMPUTATION  BY  LOGARITHMS 

357.  If  the  hypotenuse  of  a  right  triangle  and  one  leg  are  known, 
the  other  leg  may  be  found  by  means  of  logarithms,  for  if  h  =  hypo- 
tenuse, Zi  =  one  leg,  then  ^=V/i2-^i2=  V(h+li)(h-li). 

:.  log  k=$  [log  (A+w+iog  (h  -w]. 

If  the  hypotenuse  of  a  right  triangle  is  587,  and  one  leg  is  324, 
what  is  the  other  leg?  2 

358.  The  area  of  an  equilateral  triangle  whose  side  is  s,  is  7  A/3. 

(a)  Find  the  area  of  an  equilateral  triangle  whose  side  is  15.38  units. 
(6)  Find  the  side  of  an  equilateral  triangle  whose  area  is  89.5 
square  inches. 

359.  The  formula  for  the  area  of  a  triangle  in  terms  of  its  sides 
is  A  =  -\/s(s  —a)(s  —  b)  (s  —  c)  where  a,  b,  c  are  the  sides  of  the  triangle 
and  s  the  semi-perimeter.    Find  the  area  of  a  triangle  whose  sides 
are  436,  725.4,  951.8  units  respectively. 

This  is  often  referred  to  as  Heron  of  Alexandria's  formula,  and 
will  be  proved  later. 

a360.t  Show  that  the  amount  of  P  dollars  at  interest  com- 


r  Y 

r?^  )  ; 

JO/ 


pounded  annually  for  n  years  is  P  (  1  -f  ^^  )  ;  compounded  semi- 

annually  is  P  (1+— . 
\      zui 

*  It  is  suggested  that  pupils  will  find  an  inexpensive  slide  rule  of  great  use 
in  rapid  calculation.  Such  an  instrument,  called  the  "Favorite,"  can  be  pur- 
chased of  Keuffel  and  Esser,  New  York,  N.  Y. 

t  As  here,  the  letter  a  preceding  the  number  of  an  exercise  indicates  that 
algebra  beyond  the  solution  of  simple  linear  equations  is  required  to  solve  the 
problem. 


ALGEBRA  IN  APPLIED  MATHEMATICS  99 

361.  In  how  many  years  will  $1.00  double  itself  at 
(a)  3%  interest  compounded  annually? 

(6)  4%  interest  compounded  annually? 

(c)  5%  interest  compounded  annually? 

(d)  6%  interest  compounded  annually? 

362.  In  how  many  years  will  $1.00  double  itself  at 
(a)  3%  interest  compounded  semi-annually? 

(6)  4%  interest  compounded  semi-annually? 

(c)  5%  interest  compounded  semi-annually? 

(d)  6%  interest  compounded  semi-annually? 

363.  Find   the   amount   at   compound   interest,    compounded 
annually  of: 

(a)  $150  for  7  years  at  5J%."      (6)  $1850  for  5  years  at  4%. 
(c)   $10  for  50  years  at  5%. 

364.  To  find  the  present  value  A0  of  an  annuity  (a  fixed  sum  of 
money,  payable  at  equal  intervals  of  time)  of  s  dollars  to  continue 
for  n  years  at  R%  compound  interest,  the  formula 


is  used. 

Find  the  present  value  of  an  annuity  (i.e.,  the  amount  which,  if 
put  at  compound  interest  for  the  given  time  and  rate,  will  amount 
to  the  given  sum) . 

(a)  Of  $1000  for  10  years,  at  4%  compound  interest. 

(6)  Of  $1200  for  10  years,  at  4%  compound  interest. 

(c)  Of  $1500  for  10  years,  at  4%  compound  interest. 

(d)  Of  $500  for  5  years,  at  5%  compound  interest. 

365.  What  annuity  can  be  purchased  for  $3000,  if  it  is  to  run 
for  15  years,  at  5%  compound  interest  compounded  annually? 

366.  The  diameter  in  inches  of  a  connecting  rod  depends  upon 
the  diameter  D  of  the  engine  cylinder,  I  the  length  of  the  connecting 
rod,  and  P  the  maximum  steam  pressure  in  pounds  per  sq.  inch, 
according  to  Mark's  formula  d =0. 02758 VD-l\/P. 

What  is  d  when  D  =  30,  Z  =  75,  and  P=150? 

367.  If  fluid  friction  be  used  to  retard  the  motion  of  a  flywheel 
making  V0  revolutions  per  minute,  the  formula  V=V0e~kt  gives 
the  number  of  revolutions  per  minute,  after  the  friction  has  been 


100  PLANE  GEOMETRY 

applied  t  seconds.  If  the  constant  k  =  0.35,  the  value  of  e  being 
2.718,  how  long  must  the  friction  be  applied  to  reduce  the  number 
of  revolutions  from  200  to  50  per  minute? 

a368.  The  pressure,  P,  of  the  atmosphere  in  pounds  per  sq.  inch, 
at  a  height  of  z  feet,  is  given  approximately  by  the  relation  P0= 
P0e  ~  kz,  where  P0  is  the  pressure  at  sea  level  and  k  is  a  constant, 
the  value  of  e  being  2.718.  Observations  at  sea  level  give  P0  =  14.72, 
and  at  a  height  of  1122  feet,  P  =  14.11.  What  is  the  value  of  fc? 

369.  If  a  body  of  temperature  TI  be  surrounded  by  cooler  air 
of  temperature  T0 ,  the  body  will  gradually  become  cooler;  and  its 
temperature,  T,  after  a  certain  time,  say  t  minutes,  is  given  by 
Newton's  law  of  cooling,  that  is  T=T0+(Tl  -T0}e~ki,  where  k  is 
a  constant  and  e  =  2.718.    In  an  experiment  a  body  of  temperature 
55°  C.  was  left  to  itself  in  air  whose  temperature  was  15°  C.    After 
11  minutes  the  temperature  was  found  to  be  25°.    What  is  the 
value  of  &? 

370.  How  many  ciphers  are  there  between  fche  decimal  point  and 
the  first  significant  figure  in  (0.0504) 10? 

371.  The  loss  of  energy  E  through  friction  of  every  pound  of 
water  flowing  with  velocity  v  through  a  straight  circular  pipe  of 
length  I  ft.  and  diameter  d  ft.  is  given  by  0.0007z/2Z-^d. 

Given  v  =  8.5  ft.  per  sec.,  Z  =  3000  ft.,  d  =  Q  inches,  find  E. 

372.  A  man  bequeaths  $500,  which  is  to  accumulate  at  compound 
interest  until  the  interest  for  one  year  at  5%  will  amount  to  at 
least  $300,  after  which  the  yearly  interest  is  to  be  awarded  as  a 
scholarship.    How  many  years  must  elapse  before  the  scholarship 
becomes  available,  assuming  that  the  original  bequest  is  made  to 
earn  5%  compound  interest? 

373.  In  1624  the  Dutch  bought  Manhattan  Island  from  the 
Indians  for  about  $24.    Suppose  that  the  Indians  had  put  their 
money  out  at  compound  interest  at  7%,  and  had  added  the  interest 
to  the  principal  each  year,  how  large  would  be  the  accumulated 
amount  in  1910? 

Ans.  In  round  numbers  $6,000,000,000.  The  actual  valuation 
of  Manhattan  and  Bronx  real  and  personal  property  in  1908  was 
$5,235,399,980. 

374.  The  population  of  the  State  of  Washington  in  1890  was 


ALGEBRA  IN  APPLIED  MATHEMATICS  101 

349,400,  and  in  1900  it  was  518,100.  What  was  the  average  ^«?arty 
rate  of  increase?  Assuming  the  rate  of  increase  to  remain  the 
same,  what  should  have  been  the  population  in  1910? 

375.  The  founder  of  a  new  faith  makes  one  new  convert  each 
year,  and  each  new  convert  makes  another  convert  each  year,  and 
so  on.    How  long  would  it  require  to  convert  the  whole  earth  to  the 
new   faith,    assuming    that     the    population   of     the   world    is 
1,500,000,000? 

Ans.    Between  30  and  31  years. 

376.  The  combined  wealth  of  the  United  States  and  Europe  was 
estimated  (1908)  to  amount  to  about  $450,000,000,000.     Let  us 
assume  that  the  entire  wealth  of  the  world  amounts  to  $1012.    How 
long  would  it  take  $1.00  put  out  at  compound  interest  at  3%  to 
equal  or  exceed  this  amount? 

Ans.    935  years. 

The  following  problems  should  be  solved  by  means  of  five-place 
tables: 

377.  The  circumference  of  a  circle  is  2-nr  (r  being  radius) .    (Use 
7r  =  3.i416.) 

(a)  Find  the  circumference  of  a  circle  whose  radius  is  143.7. 
(6)  Find  the  radius  of  a  circle  whose  circumference  is  528.45  units. 

378.  The  area  of  a  circle  is  irr2. 

(a)  Find  the  area  of  a  circle  whose  radius  is  12.34"- 

(b)  Find  the  radius  of  a  circle  whose  area  is  243.5  sq.  ft. 

379.  The  area  of  the  surface  of  a  sphere  is  47rr2. 

(a)  The  radius  of  the  earth  is  3959  miles.    What  is  its  surface? 

(b)  What  is  the  length  of  the  equator? 

(c)  A  knot  is  the  length  of  one  degree  measured  along  the  equator. 
How  many  miles  in  a  knot? 

380.  The  volume  of  a  sphere  is  f  Trr3.    What  is  the  weight  in 
tons  of  a  solid  cast-iron  sphere  whose  radius  is  5.343  feet,  if  the 
weight  of  a  cubic  foot  of  water  is  62.355  pounds,  and  the  specific 
gravity  of  cast-iron  is  7.154? 

381.  The  stretch  of  a  brass  wire  when  a  weight  is  hung  at  its 
free  end  is  given  by  the  relation : 

=  mgl 


102  PLANE  GEOMETRY 

Where*  in  is  the  height  applied,  g  =  980,  I  is  the  length  of  the  wire, 
r  is  its  radius,  and  A:  is  a  constant.  Find  k  for  the  following  values : 
ra  =  944.2  grams,  1  =  219.2  centimeters,  r  =  0.32  centimeters,  and 
$  =  0.060  centimeters. 

382.  The  weight  P  in  pounds  which  will  crush  a  solid  cylindrical 
cast-iron  column  is  given  by  the  formula: 

,73.55 

P-98,920^, 

where  d  is  the  diameter  in  inches,  and  I  the  length  in  feet.  What 
weight  will  crush  a  cast-iron  column  6  feet  long  and  4.3  inches  in 
diameter? 

383.  The  weight  W  of  one  cubic  foot  of  saturated  steam  depends 
upon  the  pressure  in  the  boiler  according  to  the  formula : 

P0.941 

W  =  — 

330.36' 

where  P  is  the  pressure  in  pounds  per  sq.  inch.  What  is  W  if  the 
pressure  is  280  pounds  per  sq.  inch? 

384.  The  number,   n,   of  vibrations  per  second  made  by  a 
stretched  string  is  given  by  the  relation : 

fMg 
m  ' 

where  I  is  the  length  of  the  string,  M  the  weight  used  to  stretch  the 
string,  m  the  weight  of  one  centimeter  of  the  string,  and  g  =  980. 
Find  n,  when  M  =  6213.6  grams,  Z  =  84.9  centimeters,  and  m  = 
0.00670  gram. 

385.  If  p  is  the  pressure  and  u  the  volume  in  cubic  feet  of  1  Ib. 
of  steam,  then  from  ?ra1-0646=479  find  u  when  p  is  150. 

The  practical  problems  366-369/380-384,  were  taken  from  Rietz 
and  Crathorne's  College  Algebra  (Henry  Holt  and  Co.). 

The  interesting  problems  372-376  were  taken  from  White's 
Scrapbook  of  Mathematics  (Open  Court  Pub.  Co.). 

The  student  is  referred  to  such  texts  if  his  interests  or  needs 
require  further  work  in  logarithms. 


ALGEBRA  IN  APPLIED  MATHEMATICS 


103 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

11 

0414 

0453. 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

3222 

3243 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

36 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

6010 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

41 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

42 

6232 

6243 

6253 

6263 

6274 

6284 

6294 

6304 

6314 

6325 

43 

6335 

6345 

6355 

6365 

6375 

6385 

6395 

6405 

6415 

6425 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

46 

6628 

6637 

6646 

6656 

6665 

6675 

6684 

6693 

6702 

6712 

47 

6721 

6730 

6739 

6749 

6758 

6767 

6776 

6785 

6794 

6803 

48 

6812 

6821 

6830 

6839 

6848 

6857 

6866 

6875 

6884 

6893 

49 

6902 

6911 

6920 

6928 

'6937 

6946 

6955 

6964 

6972 

6981 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

52 

7160 

7168 

7177 

7185 

7193 

7202 

7210 

7218 

7226 

7235 

53 

7243 

7251 

7259 

7267 

7275 

7284 

7292 

7300 

7308 

7316 

54 

7324 

7332 

7340 

7348 

7356 

7364 

7372 

7380 

7388 

7396 

104 


PLANE  GEOMETRY 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7604 

7612 

7619 

7627 

58 

7634 

7642 

7649 

7657 

7664 

7672 

7679 

7686 

7694 

7701 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7896 

7903 

7910 

7917 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7966 

7973 

7980 

7987 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

64 

8062 

8069 

8075 

8082 

8089 

8096 

8102 

8109 

8116 

8122 

65 

8129 

8136 

8142 

8149 

8156 

8162 

8169 

8176 

8182 

8189 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

8254 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8319 

68 

8325 

8331 

8338 

8344 

8351 

8357 

8363 

8370 

8376 

8382 

69 

8388 

8395 

8401 

8407 

8414 

8420 

8426 

8432 

8439 

8445 

70 

8451 

8457 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8609 

8615 

8621 

8627 

73 

8633 

8639 

8645 

8651 

8657 

8663 

8669 

8675 

8681 

8686 

74 

8692 

8698 

8704 

8710 

8716 

8722 

8727 

8733 

8739  s 

8745 

75 

8751 

8756 

8762 

8768 

8774 

8779 

8785 

8791 

8797 

8802 

76 

8808 

8814 

8820 

8825 

8831 

8837 

8842 

8848 

8854 

8859 

77 

8865 

8871 

8876 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

81 

9085 

9090 

9096 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

82 

9138 

9143 

9149 

9154 

9159 

9165 

9170 

9175 

9180 

9186 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

84 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

85 

9294 

9299 

9304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

86 

9345 

9350 

9355 

9360 

9365 

9370 

9375 

9380 

9385 

9390 

87 

9395 

9400 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

88 

9445 

9450 

9455 

9460 

9465 

9469 

9474 

9479 

9484 

9489 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

90 

9542 

9547 

9552 

9557 

9562 

9566 

9571 

9576 

9581 

9586 

91 

9590 

9595 

9600 

9605 

9609 

9614 

9619 

9624 

9628 

9633 

92 

9638 

9643 

9647 

9652 

9657 

9661 

9666 

9671 

9675 

9680 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

95 

9777 

9782 

9786 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

96 

9823 

9827 

9832 

9836 

9841 

9845 

9850 

9854 

9859 

9863 

97 

9868 

9872 

9877 

9881 

9886 

9890 

9894 

9899 

9903 

9908 

98 

9912 

9917 

9921 

9926 

9930 

9934 

9939 

9943 

9948 

9952 

99 

9956 

9961 

9965 

9969 

9974 

9978 

9983 

9987 

9991 

9996 

ALGEBRA   IN  APPLIED  MATHEMATICS  105 

B.  RATIO,  PROPORTION  AND  VARIATION 
I.  RATIO  AND  PROPORTION 

On  page  74  a  ratio  was  defined  as  a  fraction,  and  a  fraction  as  an 
indicated  quotient.  Hence  the  ratio  of  two  numbers  is  their  indi- 
cated quotient.  We  recognize  in  this  definition  that  a  ratio  and  a 
fraction  are  identical,  i.e.,  the  ratio  of  3  to  4  and  the  fraction  f  are 
the  same.  Thus  we  see  at  the  very  beginning  that  we  are  not  deal- 
ing with  a  new  subject,  but  that,  having  learned  how  to  work  with 
fractions,  we  already  know  a  good  deal  about  ratios.  The  symbols 
for  expressing  ratio  we  are  familiar  with  as  the  symbols  for  expres- 
sing division,  viz.,  often  used  in  Europe  to  express  division,  — , 
the  fractional  form  which  we  shall  find  most  convenient,  and  shall 
use  exclusively,  and  -T-,  really  a  combination  of  the  two  symbols 
previously  given. 

The  ratio  of  concrete  numbers  can  be  found  only  when  the  quan- 
tities are  of  the  same  denomination.  When  a  common  unit  of 
measure  can  be  found  for  two  such  quantities  their  ratio  is  the 
quotient  of  the  numbers  expressing  their  measures  in  terms  of 
that  unit.  To  find  the  ratio  of  10  hours  to  5  days,  one  must  express 
both  in  terms  of  a  common  unit.  Here,  calling  5  days  120 
hours,  the  ratio  is  y1^,  or  Y1^.  It  is  not  always  possible  to  get 
the  ratio  of  two  quantities,  e.g.,  10  minutes  and  5  bushels  can 
have  no  common  measure.  Again,  it  is  not  always  possible  to 
get  an  exact  ratio,  as  in  the  case  of  the  circumference  of  a  circle 
and  its  diameter.  As  we  know,  this  ratio  is  called  TT,  and  its 
value  we  may  express  more  or  less  accurately,  but  can  never 
calculate  exactly. 

In  a  ratio  the  first  number  is  called  the  antecedent,  and  the  second 
the  consequent.  It  is  evident,  then,  that  the  antecedent  corresponds 
to  the  numerator,  and  the  consequent  to  the  denominator,  when 
we  think  of  a  ratio  as  a  fraction. 

Since  a  ratio  is  a  fraction,  one  fundamental  property  of  ratios  is 
apparent,  that  is,  both  antecedent  and  consequent  may  be  multi- 
plied or  divided  by  the  same  number  without  changing  the  value 
of  the  ratio.  How  could  you  state  this  fact  algebraically?  What 
principle  of  fractions  verifies  it? 


106  PLANE  GEOMETRY 

EXERCISES.     SET  XLII.     RATIO 

386.  What  is  the  ratio  of  3  ft.  8  in.  to  4  in.? 

387.  What  is  the  ratio  of  37i%  to  87i%? 

388.  What  is  the  ratio  of  5z2  to  (5z)2? 

389.  Simplify  the  ratio 

(s3)4 

390.  Simplify  the  ratio  ^T 

s  s  . 

391.  Simplify  the  ratio  of      -        to 


392.  Find  the  ratio  of  a  to  b  if  6a  -76  =  3a-f  46. 

393.  Does  a  ratio  always  remain  the  same  if  a  constant  is  added 
to  both  antecedent  and  consequent?    Discuss  in  detail. 

394.  What  ratio  is  implied  in  the  statement  that  the  death-rate 
of  a  certain  town  for  the  month  is  4  out  of  each  1000? 

The  statement  of  the  equality  of  two  ratios  is  called  a  proportion. 
In  other  words,  a  proportion  is  a  fractional  equation  each  member 

Cl       f* 

of  which  is  a  single  fraction  (or  ratio);  e.g.,  T)  =  ~J  is  a  proportion. 

The  first  antecedent  and  the  second  consequent  of  a  proportion  are 
called  the  extremes,  and  the  first  consequent  and  the  second  antecedent 

a     c 
are  called  the  means  of  a  proportion.     In  r  =  3,  a  and  d  are  the 

extremes,  and  6  and  c  the  means. 

The  antecedents  and  consequents  in  a  proportion  are  called  the 
terms  of  the  proportion.  //  the  first  consequent  equals  the  second 
antecedent,  each  of  those  terms  is  called  a  mean  proportional  between 
the  first  and  the  last  terms  of  the  proportion;  the  whole  expression  is 

referred  to  as  a  mean  proportion.    In  the  mean  proportion  r—-, 

C/       C 

b  is  said  to  be  a  mean  proportional  between  a  and  c.     If  we  solve 
this  equation  for  b  two  values  would  be  obtained,  ±  \/ac. 

EXERCISES.     SET  XLIIL     PROPORTION 

395.  Find  the  value  of  y  in  the  proportions 

-*  5       7 


396.  Find  the  mean  proportionals  between 

(a)  16  and  4.  (b)  -  and  ^=  (c)  a+b  and  a  -6. 


ALGEBRA  IN  APPLIED  MATHEMATICS  107 

397.  What  number  added  to  each  of  the  numbers  3,  7,  15,  and 
25  will  give  results  which  are  in  proportion? 

398.  In  sterling  silver,  the  amount  of  silver  is  .925  of  the  entire 
weight  of  the  metal,     (a)  How  many  ounces  of  pure  silver  are 
needed  to  make  500  oz.  sterling  silver?    (6)  500  oz.  of  pure  silver 
will  make  how  many  ounces  of  sterling  silver?    (Form  a  proportion 
and  solve  this  problem  by  means  of  it.) 

399.  The  volumes  of  two  similar  solids  have  the  same  ratio  as 
the  cubes  of  any  two  homologous  dimensions.     The  diameter  of 
the  first  of  two  bottles  which  have  the  same  shape  is  three  times 
the  diameter  of  the  second.    If  the  first  holds  5  ounces,  how  much 
does  the  second  hold? 

Many  properties  of  a  proportion  may  be  derived  from  its  defini- 
tion and  the  fundamental  laws  of  algebra.  We  shall  now  suggest 
proofs  for  three  theorems  of  proportion  which  are  especially  im- 
portant because  of  their  application  to  geometry. 

Theorem  28.  Any  proportion  may  be  transformed  by  alternation, 
i.e.,  the  first  term  is  to  the  third  as  the  second  is  to  the  fourth. 

n.  ^    .  a    c  To  prove:  a    b 

Given  :^-d.  -^. 

Suggestion  for  proof:  By  what  must  we  multiply  g  to  get  -  ? 

Theorem  29.  In  any  proportion,  the  terms  may  be  combined 
by  addition  (usually  called  composition)  ;  i.e.,  the  ratio  of  the  sum 
of  the  first  and  second  terms  to  the  second  term  (or  first  term) 
equals  the  ratio  of  the  sum  of  the  third  and  fourth  terms  to  the 
fourth  term  (or  third  term). 

N.B.  —  Addition  and  sum  are  used  in  the  algebraic  sense. 

Given:  £^. 

'    a±b     c±d       b±a 
To  prove:  (1)    --  -  --  or  -- 


Suggestions  for  proofs  :  (1)      ±  1  =    =*=  1.    Authority? 


a-6     c-d  b-a    d-c 

If  —  ^  —  why  does  —  =— 

Complete  the  proofs. 


108  PLANE  GEOMETRY 

Theorem  30.  In  a  series  of  equal  ratios,  the  ratio  of  the  sum  of 
any  number  of  antecedents  to  the  sum  of  their  consequents 
equals  the  ratio  of  any  antecedent  to  its  consequent. 

ace  k 

G.ven:  S-a-j-  ......  -j-  •  •  •  •  . 

a+c+e-\  ----     k 
Toprove: 


Proof:  Let^r 

.'.  a=br,  c=dr,  etc. 


Finish  the  proof. 

EXERCISES.     SET  XLIV.    APPLICATIONS  OF  PROPORTION 

400.  In  any  proportion  the  product  of  the  means  is  equal  to  the 
product  of  the  extremes. 

Hint    What  axiom  do  we  use  in  clearing  an  equation  of  fractions? 
State  a  corollary  of  this  theorem  which  will  express  the  mean 
proportional  as  a  function  of  the  other  terms  of  the  proportion. 

401.  If  the  product  of  two  numbers  is  equal  to  the  product  of 
two  other  numbers,  the  factors  of  either  product  may  be  made  the 
means  of  a  proportion  of  which  the  factors  of  the  other  are  the 
extremes. 

How  are  Exs.  400  and  401  related? 

Af\n      /    \    Tr  a        C  ^    A  a  +  &        C+d 

402.  (a)  If  T  =  3  prove  that  —  —?  =  —  —.. 

b     d.  a-b     c  -d 

(b)  State  this  fact  as  a  theorem  in  proportion  (sometimes  re- 
ferred to  as  addition  and  subtraction  or  composition  and  division). 

403.  (a)  Prove  that  if  £  =  %  then  -  =  -. 

b     d'  a      c 

(b)  State  this  fact  as  a  theorem  in  proportion.    This  transforma- 
tion is  usually  referred  to  as  inversion. 

404.  (a)  State  a  brief  way  of  testing  the   correctness   of  a 
proportion.  976     8?4 

(6)  Is  the  following  a  true  proportion    j   =  =— 


405.  A  and  B  are  in  business,  and  their  respective  shares  of  the 
business  are  in  the  ratio  of  -§-.  If  the  profits  of  a  certain  year  are 
$16000,  and  during  the  year  A  draws  $1200  and  B  $1000,  at  the 
end  of  the  year  how  much  of  the  profits  does  each  receive? 


ALGEBRA  IN  APPLIED  MATHEMATICS  109 

406.  Is  the  validity  of  a  proportion  impaired  by  adding  the  same 
number  to  all  the  terms?    Prove  your  answer  to  be  correct. 

407.  By  the  term,  specific  gravity  of  a  substance  is  meant  the  ratio 
of  the  weight  of  a  volume  of  that  substance  to  the  weight  of  an  equal 
volume  of  some  other  substance  taken  as  a  standard.    In  practice 
that  standard  is  water.    A  cubic  foot  of  water  weighs  62.4  Ibs. 

(a)  What  is  the  specific  gravity  of  steel  if  a  cubic  foot  of  it 
weighs  490  Ibs.? 

(6)  What  is  the  specific  gravity  of  ice  if  a  cubic  foot  of  it  weighs 
57.5  Ibs.? 

(c)  The  specific  gravity  of  sea  water  is  1 .024.  What  is  the  weight 
of  5  gallons? 

408.  What  is  the  weight  in  tons  of  a  solid  cast-iron  sphere  whose 
radius  is  5.433  ft.  if  the  weight  of  a  cubic  foot  of  water  is  62.355  Ibs., 
and  the  specific  gravity  of  cast-iron  is  7.154?    (See  Ex.  380.) 

409.  What  number  added  to  a  ratio  whose  antecedent  is  5  and 
consequent  is  8  will  give  the  ratio  f  ? 

410.  What  number  added  to  the  terms  of  the  ratio  T7T  will  give 
the  ratio  f  ? 

Many  of  the  important  applications  of  proportion  are  made  in 
physics.  The  exercises  in  proportion  which  follow  depend  upon 
three  well-known  facts  in  physics.  The  facts  are: 

(1)  Law  of  the  Inclined  Plane.    If  F  represents  the  force  applied 
along  an  inclined  plane,  W  the  weight  of  the  body,  h  the  height  of 
the  plane  and  /  the  length  of  the  plane,  then  ^ 

F_h       *   (jA 

TF=r 

Illustrative  Problem.     How  heavy  a  weight  could  a  force  of 
300  Ibs.  pull  up  an  incline  75  ft.  high  and  400  ft.  long? 
Substituting:  F  =  300,  ft  =  75,  2  =  400;  calculate  W. 

(2)  Boyle's  Law.    If  PI  be  the  pressure  of  a  gas  of  volume  Vi,  and 
P2  the  pressure  of  the  same  gas  of  volume  F2,  the  temperature  re- 
maining constant,  we  have     FI  _  PZ 

V*     Pi 

Illustrative  Problem.  Keeping  the  temperature  the  same,  if 
200  cu.  cm.  of  gas  exerting  a  pressure  of  2  Ibs.  per  sq.  cm.  be  allowed 
to  expand  to  a  volume  of  300  cu.  cm.  what  pressure  will  it  exert? 

Substituting:    ^  =  200,  PI  =  2,  72  =  300;  calculate  P2. 


110  PLANE  GEOMETRY 

(3)  Charles's  Law.  When  the  pressure  is  constant,  if  V\  is  the 
volume  of  a  gas  of  temperature  £i+273  centigrade  (called  absolute 
temperature)  and  V%  the  volume  of  the  same  gas  of  temperature 
fe+273,  then  V\  =  fr+273 


Illustrative  Problem.  If  200  cu.  cm.  of  gas  is  300°  C.  (absolute 
temperature)  keeping  the  pressure  the  same,  what  will  be  the 
temperature  if  the  gas  is  expanded  to  250  cu.  cm.? 

Substituting:   Fi  =  200,  F2  =  250,  *i  =  300  -273;  calculate  fe. 


EXERCISES.     SET  XLIV  (concluded) 

411.  Find  the  force  which  must  be  exerted  to  draw  a  sled  weigh- 
ing 240  Ibs.  up  a  hill  which  is  300  ft.  long  and  50  ft.  high. 

412.  A  bladder  holds  40  cu.  in.  of  air  under  a  pressure  of  15  Ibs. 
per  sq.  in.    What  is  the  size  of  the  bladder  when  the  pressure  is 
reduced  to  12  Ibs.  per  sq.  in.? 

413.  A  certain  mass  of  gas  occupying  a  volume  of  160  cu.  cm. 
at  a  temperature  of  47°  C.  is  cooled  to  17°  C.    Find  the  volume  at 
the  lower  temperature. 

414.  A  boy  is  able  to  exert  a  maximum  force  of  80  Ibs.    How 
long  an  inclined  plane  must  he  use  to  push  a  truck  weighing  320 
Ibs.  up  to  a  doorway  which  is  3  J  ft.  above  the  level  of  the  ground? 

II.  VARIATION 

//  one  variable  is  a  constant  number  of  times  a  second  variable, 
the  first  quantity  is  said  to  vary  as  the  second.  If  x=ky,  then  x 
is  said  to  vary  as  y,  and  is  written  x  oc  y.  The  circumference  of  a 
circle  varies  as  its  diameter  because  it  is  equal  to  a  constant  (IT) 
times  its  diameter,  i.e.  C=7rD  or  CccD. 

The  illustration  cited  is  called  direct  variation,  but  often  two 
quantities  are  said  to  vary  inversely  when  an  increase  in  the  one 
causes  a  proportional  decrease  in  the  other,  e.g.,  the  time  it  takes  to 
go  from  America  to  Europe  varies  inversely  as  the  speed  of  the 

vessel.  If  we  call  the  time  t  and  the  speed  s  we  might  write  tcc-gt 
Again,  we  might  illustrate  this  inverse  variation  by  the  apparent 
height  of  objects  and  our  distance  from  them.  If  a  building 
appears  6"  high  when  we  are  200  feet  from  it,  how  high  will  it 
appear  when  we  are  only  100  feet  from  it? 


ALGEBRA  IN  APPLIED  MATHEMATICS  111 

A  quantity  is  said  to  vary  jointly  as  several  others  if  it  is  equal  to 
a  constant  number  of  times  the  product  of  the  others;  i.e.,  a  varies 
jointly  as  c,  d,  and  e  (accede),  if  a=kcde.  This  relation  may  be 
illustrated  by  the  area  of  a  triangle  which  varies  jointly  as  the  base 
and  altitude.  In  this  case  &  =  i.  Again,  we  might  illustrate  this 
relation  by  a  man's  wages  which  vary  jointly  as  the  number  of 
days  he  works  and  the  pay  he  receives  for  one  day's  work.  The 
constant  in  this  as  in  many  other  cases  is  unity. 

EXERCISES.     SET  XLV.    APPLICATION  OF  VARIATION 

415.  The  velocity  of  a  falling  body  varies  directly  as  the  time 
during  which  it  falls,     (a)  State  this  fact  as  a  formula.     (6)  If 
the  velocity  of  a  body  is  160  feet  per  second  after  falling  5  seconds, 
what  will  the  velocity  be  after  12  seconds? 

Notes  on  Solution:    Such  problems  may  be  solved  by  first  solving  for  the 
constant  or  by  throwing  them  into  the  form  of  a  proportion. 

Here:  By  the  first  method  160  =  5fc  .'.  &  =  32  /.  i>  =  32- 12  =  384, 

5        12  12.160 

or  by  the  second  method  -^  =  -      .*.  v  —  — = —  =384. 

loU       v  o 

(c)  How  long  will  a  body  fall  before  acquiring  a  velocity  of 
520  feet  per  second? 

416.  The  distance  through  which  a  body  falls  from  rest  varies 
as  the  square  of  the  time  during  which  it  falls. 

(a)  State  this  fact  as  a  formula. 

(6)  If  a  body  falls  576  ft.  in  6  sees.,  how  far  does  it  fall  in  10  sees.? 

(c)  How  far  will  a  body  fall  in  12  seconds? 

(d)  How  far  will  a  body  fall  during  the  twelfth  second? 

(e)  How  long  will  it  take  a  body  to  fall  a  mile? 

417.  The  pressure  of  wind  on  a  flat  surface  varies  jointly  as  the 
area  of  the  surface  and  the  square  of  the  wind's  velocity. 

(a)  State  this  fact  as  a  formula. 

(b)  The  pressure  of  the  wind  on  1  sq.  ft.  is  0.9  Ib.  when  the 
velocity  of  the  wind  is  15  miles  per  hour.    What  is  the  pressure  of 
the  wind  against  the  side  of  a  house  120  feet  deep  and  70  feet  high 
when  the  wind  is  blowing  40  miles  an  hour? 

(c)  What  is  the  pressure  on  the  same  house  when  the  wind  is 
blowing  60  miles  per  hour? 


112  PLANE  GEOMETRY 

418.  The  heat  one  derives  from  a  stove  varies  approximately 
inversely  as  the  square  of  one's  distance  from  the  stove.  If  I  move 
my  position  from  10  feet  away  from  a  stove  to  35  feet  away,  what 
part  of  the  original  heat  will  I  then  receive? 

419.  The  law  of  gravitation  states  that  the  weight  of  a  body  varies 
inversely  as  the  square  of  its  distance  from  the  center  of  the  earth. 

(a)  If  a  body  weighs  10  Ibs.  on  the  surface  of  the  earth  what 
will  it  weigh  5  miles  above  the  surface?  (Consider  the  radius  of 
the  earth  to  be  4000  miles.) 

(6)  How  high  would  a  body  have  to  be  raised  above  the  surface 
of  the  earth  to  lose  half  its  weight? 

420.  The  intensity  of  light  varies  inversely  as  the  square  of  the 
distance  from  its  source. 

(a)  How  much  farther  from  a  lamp  20  feet  away  must  a  piece 
of  paper  be  moved  to  receive  half  as  much  light? 

(6)  What  is  the  relation  of  the  intensity  of  light  15  feet  from  an 
electric  light  and  37  feet  from  the  same  light. 

1421.  Kepler  proved  that  the  squares  of  the  times  of  revolution 
of  the  planets  about  the  sun  vary  as  the  cubes  of  their  distances 
from  the  sun.  The  earth  is  93,000,000  miles  from  the  sun,  and 
makes  a  revolution  in  approximately  365  days.  How  far  is  Venus 
from  the  sun  if  it  makes  one  revolution  in  226  days?  (Use  logs.) 

422.  The  strings  of  a  musical  instrument  produce  sounds  by 
vibrating.    The  number  of  vibrations  in  any  fixed  interval  of  time 
varies  directly  as  the  length  of  the  string,  if  the  strings  are  alike 
in  other  particulars. 

A  C  string  42"  long  vibrates  256  times  per  second.  A  G  string, 
like  the  C  string  except  for  length,  vibrates  384  times  per  second. 
How  long  must  it  be? 

423.  The  relation  between  the  time  of  oscillation  of  a  pendulum 
and  its  length  is  given  by  the  following  formula: 

If  two  pendulums  are  of  lengths  L  and  I  respectively  and  the 
number  of  oscillations  per  second  are  T  and  t  respectively,  then: 

T2     L 

¥^J' 

(a)  A  pendulum  which  makes  1  oscillation  per  second  is  39. 1* 
long.  How  often  will  a  pendulum  156.4"  long  vibrate  per  second? 


ALGEBRA  IN  APPLIED  MATHEMATICS 


113 


(6)  How  long  would  a  pendulum  have  to  be  to  oscillate  once  a 
minute? 

1424.  The  relation  between  Q,  the  quantity  of  water  in  cubic 
feet  per  second  passing  over  a  triangular  gauge  notch,  and  H,  the 
height  in  feet  of  the  surface  of  the  water  above  the  bottom  of  the 


notch,  is  given  by 


Q  ac 


When  H  is  1,  Q  is  found  to  be  2.634.  What  is  the  value  of  Q 
when  H  is  4? 

If  the  area  of  the  reservoir  supplying  the  notch  is  80000  sq.  ft., 
find  the  time  in  which  a  volume  of  water  80000  sq.  ft.  in  area  and 
3  inches  in  depth  will  be  drawn  off  when  H  remains  constant  and 
equal  to  4  ft. 

(The  relation  between  Q  and  H  may  be  written  Q=kH^  where 
k  is  a  constant.) 

425.  In  steam  vessels  of  the  same  kind  it  is  found  that  the 
relation  between  H,  the  horse-power;  V,  the  speed  in  knots;  and 
D,  the  displacement  in  tons,  is  given  by 

HocVD*. 

Given  H  =  35640,  7  =  23,  and  D  =  23000,  find  the  probable 
numerical  value  of  H  when  V  is  24. 

426.  Some  particulars  of  steam  vessels  are  given.  Assuming  in 
each  case  the  relation  H.P.  oc  V3D%  to  hold,  where  H.P.  denoted  the 
horse-power  at  a  speed  of  V  knots  and  displacement  D  in  tons, 
find  in  each  case  the  probable  H.  P.  necessary  to  give  the  indicated 
speed. 


Name 

H.  P. 

V 

D 

(i)  Paris  

20000 

20.25 

15000 

(ii)  Teutonic  

19.50 

13800 

(iii)  Campania  

22.10 

19000 

(iv)  Kaiser  

22.62 

20000 

(v)  Oceanic  



20.50 

28500 

(vi)  Communipaw  

23.00 

23000 

114  PLANE  GEOMETRY 

g427.*  Assuming  that  the  circumference  of  a  circle  is  3|  times 
its  diameter,  make  a  graph  showing  that  the  circumference  varies 
as  the  diameter. 

Some  of  the  numerous  applications  of  Ratio,  Proportion,  and 
Variation  to  geometry  will  be  given  or  suggested  in  the  pages  of 
this  book. 

LIST  OF  WORDS  DEFINED  IN  CHAPTER  V 

Logarithm,  base,  characteristic,  mantissa,  interpolation,  antilogarithm. 
Ratio,  proportion,  antecedent,  consequent,  terms,  extremes,  means,  mean 
proportional,  mean  proportion,  addition  or  composition,  alternation,  subtrac- 
tion or  division,  inversion.  Variation,  direct,  inverse,  joint. 

SUMMARY  OF  THEOREMS  PROVED  IN  CHAPTER  V 

28.  Any  proportion  may  be  transformed  by  alternation. 

29.  In  any  proportion  the  terms  may  be  combined  by  addition. 

30.  In  a  series  of  equal  ratios,  the  ratio  of  the  sum  of  any  number  of  ante- 
cedents to  the  sum  of  their  consequents  equals  the  ratio  of  any  antecedent 
to  its  consequent. 

*  As  here,  "g"  preceding  the  number  of  an  exercise  indicates  that  its  solu- 
tion involves  a  knowledge  of  graphs. 


CHAPTER  VI 


SIMILARITY 

A.  INTRODUCTORY  THEOREMS 
EXPERIMENT  I 

1.  Construct  a  scalene  triangle. 

(a)  Divide  one  side  into  4  equal  sects,  and  through  the  first 
point  of  division  construct  a  line  parallel  to  a  second  side  of  the 
triangle.  Compare  the  lengths  of  the  sects  thus  cut  on  the  third 
side  of  the  triangle. 

(6)  Repeat  the  work  of  (a) ,  dividing  the  first  side  of  the  triangle 
into  5  equal  sects. 

(c)  Repeat  the  work  of  (a)',  dividing  the  first  side  of  the  triangle 
into  9  equal  parts  and  drawing  the  line  through  the  fourth  point 
of  division. 

2.  Repeat  the  work  of  1,  using  an  equilateral  triangle. 
Theorem  31.   A  line  parallel  to  one  side  of  a  triangle,  and 

cutting  the  other  sides,  divides 
them  proportionally. 

(External  division  of  the 
sides  will  be  considered  later.) 
Given:  A  ABC,  D  in  ~BC  and  E  in 

AB,  so  that  I)E  \\  CA. 

AE    CD 
Prove:        ^ 


(D 


ADEA 


EZ 

EB' 


(2)  Similarly  it  can  be  shown  that 
A  EDC  =  CD 

"EB: 

h-DE  .  ,    . 
— — ,  h  being 


PROOF 

(1)  Triangles  having  one  dimen- 
sion equal  compare  as  their  remain- 
ing dimensions. 


ADBE 

(3)  But  if  A  DCE 


the  distance   between  DE  and  CA, 


(3)  Data  and 
equidistant. 


Is  are  everywhere 


115 


116 


PLANE  GEOMETRY 


(4)  and 


(5) 


A  PEA  _  A  EDC 
A  DBE~  ADBE' 


/f*\    •  •L/      

(o;  .  .-==7=  __ 


(4)  Quantities  equal  to  the  same 
quantity  equal  each  other. 

(5)  Quotients  of  equals  divided  by 
equals  are  equal. 

(6)  See  (4). 


EA  =  (JD 
EB~  DB 

Cor.  1.  One  side  of  a  triangle  is  to  either  of  the  sects  cut  off 
by  a  line  parallel  to  a  second  side,  as  the  third  side  is 
to  its  homologous  sect. 


AE 


CD 


Suggestion:  If  •=•  =  ^=-  why  would 
EB        DB 


~AE+EB      CD  +  DB 


\AA,/ 

/y 

x/  BI/\B 

Y/C//\C 
*-/  ^i/   /      \G 

2 

/z  V 

/  w 
Cor.  3. 

\ 

Parallels  i 

EB  DB 

Cor.  2.  A  series  of  parallels  cuts  off 
proportional  sects  on  all  trans- 
versals. 

AB      BC      ,'BC      ~CD    - 


=CiA,  if  AXY  and  BZW  are 
how  drawn? 

Parallels  which  intercept  equal  sects  on  one  trans- 
versal, do  so  on  all  transversals. 

Cor.  4.  A  line  which  bisects  one  side  of 
triangle,  and  is  parallel  to  the 
second,  bisects  the  third. 

EXERCISES.     SET  XLVI.     PROPORTIONAL  SECTS 

428.  A  sheet  of  ruled  paper  is  useful  in  dividing  a  given  sect  into 
equal  parts,  (a)  Explain.  (6)  Make  such  an  instrument  by  using 
a  sheet  of  tracing  paper  and  drawing  at  least  twenty  parallels. 
(What  must  you  be  sure  that  these  parallels  do?) 

429.  Draughtsmen 
and  designers  sometimes 
divide  a  given  sect  into 
any  required  number  of 
equal  parts  by  the  fol- 
lowing method:  To  di- 
vide AB  into  5  equal 


SIMILARITY 


117 


parts,  draw  AC  at  any  convenient  angle  with  A  B.  Draw  BD 
parallel  to  AC.  Beginning  at  A,  mark  off  on  AC  five  equal  sects 
a,  b,  c,  etc.,  of  any  convenient  length.  Beginning  at  B,  mark  off 
on  BD  five  sects  equal  in  length  to  those  on  AC,  a\t  bi,  c\,  etc. 
Join  their  extremities  as  in  diagram.  These  lines  divide  A  B 
into  5  equal  sects.  Prove  that  this  is  a  correct  method. 

430.  Among  the  applications  of  the  propositions  on  parallel 
lines  is  an  interesting  one  due  to  Arab  Al-Nairizi  (ca.  900  A.  D.). 
The  problem  is  to  divide  a  sect  into  any  number  of  equal  parts. 
He  begins  with  the  case  of  trisecting  a  sect  AB. 

Make  BQ  and  AQi  perpendicular  to  AB,  and  make  BP=PQ  = 

Q  APi  =  PiQi.  Then  A  XYZ  is  con- 
gruent to  A  YBP,  and  also  to 
:APi.  Therefore  AX=XY  = 
YB.  In  the  same  way  we  might 
continue  to  produce  BP,  until  it 
is  made  up  of  n  lengths  BP,  and 
so  for  A  PI,  and  by  properly  join- 
ing points  we  could  divide  AB  into  n-f-1  equal  parts.  In  par- 
ticular, if  we  join  P  and  PI,  we  bisect^  the  sect  AB.  Prove  the 
truth  of  these  statements. 

Divide   a  sect  into  seven  equal 
parts  by  this  method. 

431.  Find  the  cost  of  fencing  the 
field  represented  in  the  diagram. 
Field  is  drawn  to  scale  indicated, 

and  the  fence  costs  $2.75  per  rod.   °         50        100  200  rods 

432.  If  DE  is  parallel  to  BC  in  triangle  ABC,  compute  the  sects 
left  blank  from  those  given  in  the  following  table: 


AD 

DB 

AE 

EC 

AB 

AC 

24 
14 

27 

28 
56 
112 

18 

12 

18 

42 
418 
342 

433.  Divide  a  sect  11  units  long  into  parts  proportional  to  3, 
5,  7,  and  9. 


118  PLANE  GEOMETRY 

First  compute  the  lengths  of  the  required  sects,  then  construct 
them,  and  measure  the  sects  so  obtained.  Compare  the  results. 
Which  method  is  more  convenient?  Which  is  more  accurate? 

434.  The  accompanying 
diagram  suggests  a  method 
for  finding  the  distance  from 
a  point  A  to  a  second  point 
B,  visible  from  but  inacces- 
sible to  the  first  point  A . 
Hints:  C  is  selected,  from  which  both  A  and  B  are  visible.  CB  \\ED. 

435.  If  D^is  parallel  to  £C  in  triangle ,4  5C,  prove  that  =  =  =. 

AE     EC 

,ioc    TT  j     +u  j-+-  4.1,  +AD+AE     AD 

436.  Under  the  same  conditions  show  that — =  =  — - 

437.  State  Exs.  435^36  in  words. 

EXPERIMENT  H 

1.  Construct  a  scalene  triangle. 

(a)  Divide  two  sides  of  the  triangle  into  8  equal  sects.  Join 
the  corresponding  points  of  division.  By  comparing  certain  angles, 
establish  a  relation  between  the  lines  just  drawn  and  the  third 
side  of  the  triangle. 

(6)  Repeat  (a),  dividing  the  sides  into  7  equal  sects. 

2.  Repeat  1,  using  an  equilateral  triangle. 

Theorem  32.    A  line  dividing  two  sides  of  a  triangle  proportion- 
ally is  parallel  to  the  third  side.      B 
UD 

Given:   A  ABC  and  ==, 

E 

,  V.A/m  JDU,  &  m  a  Kt  ' 

EA 

To  prove:   DE  \\  CA. 

Proof:  Details  to  be  sup-    /  J^sj^. 

plied  by  the  student.  A 

T~\  r>          "r>  77T 

Draw  CY  \\  DE  cutting  BA  in  X.  Then  —  =  ==.     Why? 

Show  why  X  must  coincide  with  A. 

Cor.  1.    A  line  dividing  two  sides  of  a  triangle  so  that  those 

sides  bear  the  same  ratio  to  a  pair  of  homologous  sects 

is  parallel  to  the  third  side. 


SIMILARITY 


B.  IDEA  OF  SIMILARITY 


119 


Earlier  in  the  book  it  has  been  noted  that  two  figures  are  called 
similar  if  they  have  the  same  shape.  The  symbol  (c/3),  due  to 
Leibnitz,  for  "is  (or  are)  similar  to,"  it  has  been  pointed  out,  is 
an  S  thrown  on  its  side.  The  S  was  doubtless  used  because  it  is 
the  initial  letter  of  the  word  "similis"  (Latin  for  "like").  Before 
developing  the  subject  we  need,  however,  a  more  careful  definition 
of  similarity,  for  shape  is  only  a  vague  notion  and  not  a  scien- 
tifically defined  term. 

Before  defining  similar  figures  we  must  note  the  meaning  of 
similar  sets  or  systems  of  points.  The  points  Ai,  BI,  Ci,  ....  and 
Az,  B2)  C2,  ....  are  said  to  be  similar  systems  if  they  can  be  so 
placed  that  all  the  sects  joining  corresponding  points,  AiA*,  BiB2, 
CiCz,  . . . . ,  pass  through  the  same  point,  and  are  divided  by  that 
point  into  sects  having  the  same  ratio. 


Flo.  1. 


FIG.  2. 


In  Figs.  1  and  2,  points  A1}  Bi}  Ci,  DI  and  A2,  B2,  C2,  D2  are 
similar  systems.   A\AZ,  BiB2,  CiC2,  and  DiD2  pass  through  P  and 

An          p>  ~p         r~i  "p          JT)P 

=     =     -     -  r-  In  Fig-  2  where  p  lies  on  the 


longation  of  sects  AiAz,  BiB2,  etc.,  it  is  said  to  divide  these  sects 
externally.  The  topic  "External  Division  of  a  Sect"  will  be  further 
developed  in  the  "Second  Study." 

The  point  P  is  called  the  center  of  similitude,  and  the  ratio  r  is 
called  the  ratio  of  similitude. 

Similar  figures  are  those  which  can  be  placed  so  as  to  have  a  center 
of  similitude. 


120 


PLANE  GEOMETRY 


The  following  are  illustrations  of  similar  figures : 
^  A, 


(a)  Triangles. 


^SIB* 


(c)  Circles.  ,  (d)  General  Curvilinear 

We  are  now  in  a  position  to  prove  our  right  to  the  use  of  the 
double  symbol  ( ^  )  for  congruence. 

Cor.    Congruent  figures  are  similar. 

If  n-gon  ABC  ....  n^AiBiCi  .  ...  HI,  they  may  be  made 
to  coincide.     Then  any  point  O  may  be  taken  as  the  center  of 


OA       OB 
similitude,  and  ==  = 


-i 


SIMILARITY 


121 


EXERCISES.     SET  XLVII.     MEANING  OF  SIMILARITY 

438.  In  (a)  what  is  the  ratio  of  similitude?    What  is  the  center 
of  similitude?     If  ~OAi  =  A^A~2  and  OB*  =  0.5",  what  does  ~BJ1 
equal? 

439.  In  (a)  draw  a  sect  through  0  which  will  be  divided  by  sides 
of  triangles  in  the  ratio  of  OAi  to  OA2.     How  many  such  lines  can 
be  drawn? 

440.  In  (c)  if  0#i  =  5  cm.  and  0#2  =  1  dm.  what  is  the  ratio  of 
OAi  to  OA£     Can  you  mention  any  other  sects  in  this   figure 
having  the  same  ratio? 

441.  In  (d)  if  OCi  is  2  of  dC2  what  is  the  ratio  of  similitude? 
Under  the  same  conditions  what  is  OA2  if  OA i  is  f  "1 


Frangois  Vieta 

Shadows  furnish  familiar  illustrations  of  similar  figures, 
such  cases  the  source  of  light  is  the  center  of  similitude. 


In 


R6n6  Descartes 


The  lens  of  the  camera  gives  a  figure  similar  to  the  object  in 
front  of  it  with  the  image  inverted. 


122 


PLANE  GEOMETRY 


In  reducing  or  enlarging  maps  we  have  another  familiar  illus- 
tration of  the  application  of  the  principle  of  similarity. 


EXERCISES.     SET  XLVII  (concluded) 

442.  State  any  illustrations  or  applications  of  similarity  with 
which  you  are  familiar. 

443.  If  the  ratio  of  similitude  is  1,  what  relation  between  the 
figures  exists  besides  similarity. 

C.  SIMILARITY  OF  TRIANGLES 

*  Theorem  33.  The  homologous  angles  of  similar 
\  triangles  are  equal,  and   the  homologous  sides 

have    a    constant 
ratio. 

Given  :   A  XYZ  ™  A 

ABC. 

Prove:  (I)  ^X  =  %.A, 

XrY^^B,   ^Z^Z-C. 

..  XY      YZ     ZX 


SIMILARITY 


123 


PROOF 


(1)  .'.AXYZ  ~  A  ABC,  it  may  be 
placed  in  the  position  of  £±X\Y\Z]_ 
with  O  a  center  of  similitude  of  it 
and  A  ABC. 

(2)  :.OXi^OTi=OZi 

'OA~~OB'~OC 

(3)  :. 


CA 


(4)  /. 


4.  OBC  and 


(1)  Data  and  def.  of  sim.  figs. 

(2)  Def.  of  center  of  similitude. 

(3)  Why? 

(4)  Why? 

(5)  Why? 


(6)  Why? 

(7)  Why? 

(8)  Why? 


(5)  .'.  What  two  3*  of  A  XiY^ 
or  A  X  YZ  are  equal  to  what  two  3? 
of  A  ABC? 

(6)  Are  the  third  2£  equal? 

(7)  V  %-X  =  ^A  how  can  A  XYZ 
be  placed  with  respect  to  A  ABC? 

(8)  What    proportion    will    this 
give  us? 

(9)  In  how  many  ways  will  you 
have  to  superpose  A  XYZ  in  order 
to  prove  (II)? 

Complete  the  proof. 

^Theorem  34.    Triangles  are  similar  when  two  angles  of  one  are 
equal  each  to  each  to  two  angles  of  another. 


Given:    ^.P  =  ^.A,  ^.Q  =  ^.B  in 

&PQS  and  ABC. 
Prove:  A  PQS  ™  A  ABC. 
Suggestions  for  proof: 

Draw  PiQi  1  1  AB  and  equal 
to  PQ.      _          _ 

Draw  APiX  and  BQiZ. 

(1)  If  AJ\X  \\BQZ,  then  ABQ 
is  a  O. 

(2)  andAB=PQ 

(3)  .'.  APQS  *  AABC. 

If  APiX  intersects  ~BQZ  at  0, 
draw  PiR  \\  AC  and  intersecting  CO 
at  R.  Draw  QiR. 


*S 


(1)  Why? 

(2)  Why? 

(3)  Why? 


124 


PLANE  GEOMETRY 


...  BO     AO      . 

(4)  Thenand 


(6)  Any  sect  through  O  cut- 
ting  PiQi  in  V  and  A£  in  T7  is 

TO    AO 
divided  so  that         s~ 


(4)  Why? 

(5)  Why? 

(6)  Why? 

(7)  Why? 

(8)  Why? 

(9)  Why? 

(10)  Def.  of  congruent  figures. 

(11)  Def.  of  similar  figures. 


(7)  :. 

(8)  /. 

(9)  /. 

(10)  .'.  APQS  may  be  made  to 
coincide  with  APiQi-R. 

(11)  .'.  A  PQS  "  AABC. 

Discussion :  Consider  the  instance  in  which  PQ=AB.  Note  that  the  figures 
may  be  so  placed  that  the  center  of  similitude  lies  between  them.  Is  a  proof 
necessary  for  this  case? 

EXERCISES.     SET  XLV1II.     SIMILARITY  OF  TRIANGLES 

444.  State  a  necessary  and  sufficient  condition  for  the  similarity 
of  (a)  right  triangles,   (b)  isosceles  triangles,  and   (c)   equilateral 

triangles. 

445.  If  rays  of  light  from 
a  tree  (TTi)  pass  through  a 
hole  (H)  in  a  fence  (F)  and 
strike  a  wall,  an  inverted 
outline  (0\O)  of  the  tree  will 
be  seen  on  the  wall. 

(a)  Explain  why  this 
should  be. 

(6)  If  the  distance  HD  is 
35  ft.  and  HP  is  9  ft,  and  the  height  of  the  image  (O^O)  is  8'  8", 
find  the  height  of  the  tree. 

(c)  Under  what  conditions,  if  ever,  will  the  image  be  the  height 
of  the  tree? 

d446.  The  location  of  the  image  AI  of  a  point  A,  formed  in  a 
photographer's  camera,  is  approximately  found  by  drawing  a 


SIMILARITY 


125 


D 


straight  line  AAi  through  the 
center  of  the  lens  L.  If  CE  is 
the  position  of  the  photographic 
plate,  then  AiBi  is  the  image  of 
AB.  How  large  is  AiBi  if  AB  =  6 
ft.,  LD=12  ft.,  and  LF  =  Q  in.? 

447.  Show  that  an  object  which 
appears  of  a  certain  height,  will, 
when  moved  twice  as  far  away, 
appear  to    be  comparatively  of 
only  one-half  the  height. 

Hint:   Show  that  XB  =  %AB. 

448.  To  measure  indirectly  from  an  accessible  point  A  to  an 
inaccessible  point  B}  construct  AD  perpendicular  to  the  line  of 

sight  from  A  to  B,  and  ED  perpendi- 
cular to  AD.  Let  C  be  the  point  on 
AD  which  lies  in  line  with  E  and  B. 
By  measurement,  ED  is  100  ft.,  CD 
90  ft.,  and  CA  210  ft.  What  is  the 
distance  across  the  river? 

449.  A  man  is  riding  in  an  automobile  at  the  uniform  rate  of 
30  miles  an  hour  on  one  side  of  a  road,  while  on  a  footpath  on  the 
other  side  a  man  is  walking  in  the  opposite  direction.    If  the  dis- 
tance between  the  footpath  and  the  auto  track  is  44  ft.,  and  a  tree 
4  ft.  from  the  footpath  continually  hides  the  chauffeur  from  the 
pedestrian,  does  the  pedestrian  walk  at  a  uniform  rate?     If  so, 
at  what  rate  does  he  walk? 

450.  A  mirror  (referred  to  as  a  "speculum")  has  been  used 
for  crudely  measuring  the  height  of  objects,  such  as  trees. 

In  the  diagram  a  mirror  is  placed  hori-  G 

zontally  on  the  ground  at  M.  The  obser- 
ver takes  such  a  position  that  the  top  of 
the  tree  (C)  is  visible  in  the  mirror.  What 
distances  must  he  measure  to  be  able  to 
compute  the  height  of  the  tree  (B(?)? 

NOTE. — Light  is  reflected  from  the  surface  of  a 


X      M 

mirror  at  an  angle  equal  to  the  angle  at  which  it  strikes  it. 


126 


PLANE  GEOMETRY 


451.  The  accompanying  diagrams  show  a  simple  device  for 
measuring  heights,  using  a  square  (such  as  ABCD),  with  a  plumb- 
line  (AE)  suspended  from  one  corner.  BC  and  DC  are  divided 
into  equal  parts  (say  10,  100,  or  1000). 


Study  the  diagrams,  and  show  how  to  use  the  square  in  each  case. 

452.  Look  up  a  description  of  the  hypsometer,  and  construct  one 
of  wood,  stiff  pasteboard,  or  any  material  that  can  be  used  prac- 
tically. 

(See  D.  E.  Smith,  "  Teaching  of  Geometry."    Ginn  &  Company.) 

453.  The  distance  from  an 
accessible  point  B  to  an  in- 
accessible one,  .4,  was  meas- 
ured in  the  sixteenth  century 
by  the  use  of  drumheads.  On 
a  drumhead  placed  at  B  a 
sect  I  was  drawn  toward  A, 

and  another  m  toward  an  accessible  point  C.    BC  was*  measured. 
The  drumhead  was  then  placed  at  C  with  m  in  the  direction  CB. 

/YYl 

The  ratio  of  ~~  was  noted.    A  sect  p  was  drawn  in  the  direction 

CA.    Would  any  further  measurements  be  necessa£y  to  make  it 

possible  to  compute  AB1    Explain. 
454.  To  a  convenient  scale  draw 
.N"    a  symmetrical  roof,  pitch  7  inches 
to  the  foot,  on  MN,  which  is  to  repre- 
sent 30'  6".    The  figure  suggests  the 
construction. 


SIMILARITY 


127 


455.  Another  way  of  determining  the  distance  from  A  to  an 
inaccessible  point  B  is  to  align  A,  D,  and  B.    Run  DE  at  random. 
Run^LC  parallel  to  DE.    Align  C, 

E,  and  B.  Measure  off  FA  equal 
to  ED.  Measure  CFy  EF,  CA. 
Show  how  to  compute  AB  and 
justify  the  method. 

456.  The  accompanying  diagram  J-  —  ^  —  -4 
shows   how    coordinate  (squared) 

paper  may  be  used  to  divide  a  given  sect  into  a  number_of  equal 

parts  (here  9).  If  the  sect  AB  is  laid 
off  on  one  of  the  horizontal  lines,  the 
vertical  lines  will  be  perpendicular  to  it 
(CB-LAB).  Draw  AC.  At  the  first 
division  on  CB  from  C  draw  FE  \\  BA. 
(a)  Why,  then,  is 


c 


EF  =       ?  (6)  What  part  of  ~AB  is  JVM? 
y 

(c)  Can  you   find   a   sect   equal  to  f 
(AB)  in  the  diagram? 
457.  The  principle  of  the  diagonal  scale  is  the  same  as  that 
underlying  the  division  of  a  sect  by  the  method  of  Ex.  456.    In  the 
accompanying  diagram  of  a  diagonal  scale  the  unit  is  marked  u. 
What  is  the  length  ofAB,CD,  W,  and  MF1 

u 


p 

I9 

I 

1 

1  8 

7 

1 

6 

n 

5 

A/ 

P 

k 

DJ 

K 

2 

B 

1  1 

1  1 

BIO 

A 

10  9876543210                                    1                                     2                                    3 

458.  Show  how  by  the  use  of  the  diagonal  scale  to  measure  0.3w, 
0.56w,  0.75w,  1.8w,  Z.lu,  0.35^,  0.82^,  2.67u. 

459.  Draw  a  triangle  and  measure  the  lengths  of  the  sides  to 
hundred ths  of  an  inch  by  the  use  of  a  diagonal  scale  in  which  u  =  1 
inch.    In  measuring  adjust  the  dividers  to  the  side  of  the  triangle, 
then  apply  them  to  the  diagonal  scale. 

460.  Make  a  diagonal  scale  on  tracing  paper,  or  of  stiff  card- 
board or  of  wood,  and  with  its  aid  measure  correct  to  .01  in.   (a) 


128 


PLANE  GEOMETRY 


the  hypotenuse  of  a  right  triangle  whose  legs  are  1  in.  and  2  in.; 

(6)  the  diagonal  of  a  square  of  side  1  in.;  (c)  the  altitude  of  an  equi- 

lateral triangle  of  side  2  in.;  (d)  Verify  each  measurement  by  com- 

putation, assuming  the  Pythagorean  Theorem. 

d461.  To  find  the  height  of  an  object  AB:  Place  the  rod  CD  in 

A.  an  upright  position.  Stand 
at  X  and  sight  over  D  to  A. 
Move  the  rod  any  conve- 
nient distance,  so  that  it 
takes  the  position  Did,  and 


D  ^-- -"""A^"" 

CE      It 


C 


X1C1 


sight  over  Dl  to  A.    What 
how  may  the  height  of  the 


measurements   are  necessary,  and 
object  be  determined  from  them? 

Theorem  35.  Triangles  which  have  two  sides  of  one  propor- 
tional to  two  sides  of  another  and  their  included  angles  equal 
are  similar.  £ 

Given:  A^BCandA 


. 
AC       EC 

To  prove:  A  ABC  <» 

A  AiBiCi. 
Suggestions  for  proof: 

Place  AAiBiCi  in  position  of  A2B2C.     How  do  you  know  this  is  possible? 

Why  will  AtB2  be  parallel  to  ZB? 

What  foUows  about  %.B2A2C  and  ^.BACl 

Can  you  throw  this  theorem  back  to  the  one  immediately  preceding? 

EXERCISES.     Set  XLVIII  (continued) 

462.  Extend  your  arm  and  point  to  a  distant  object,  closing 
your  left  eye  and  sighting  across  your  finger  tip  with  your  right 
eye.  Now  keep  your  finger  in  the  same  position  and  sight  with 
your  left  eye.  The  finger  will  then  seem  to  be  pointing  to  an  object 
some  distance  to  the  right  of  the  one  at  which  you  were  pointing. 
If  you  can  estimate  the  distance  between  these  two  objects,  which 
can  often  be  done  with  a  fair  degree  of  accuracy,  especially  when 
there  are  buildings  of  which  we  can  judge  the  width  intervening, 
then  you  will  be  able  to  tell  approximately  the  distance  of  your 
finger  from  the  objects  by  the  distance  between  the  objects,  for 
it  will  be  ten  times  the  latter.  Find  the  reason  for  this. 


SIMILARITY 


129 


463.  Explain  how  the  accom- 
panying figure  can  be  used  to 
find  the  distance  from  A  to  B 
on  opposite  sides  of  a  hill. 
CE=\BC,  CD=\AC.  El)  is  found  by  measurement  to  be  125  ft. 
What  is  the  distance  AB? 

464.  The  accompanying  picture  shows  a  pair  of  proportional 
compasses.  Note  that  rods  A  B  and  CD  are  of  equal  length 
and  pivoted  together  at  0. 
(a)  Prove  AAOC&"  BOD. 

/T\  T-»         x    a 
(6)  Prove -=5. 

(c)  How  may  such  an  instrument  be  used  to  divide  a  sect? 

(d)  How  may   such  an   instrument  be  used  to   con- 

struct a  triangle  similar  to  a  given 
triangle? 

c*465.  This  picture  shows  a  pair 
of  sector  compasses.  It  can  be  used  in 
much  the  same  way  as  the  proportional 
compasses.  Show  how  by  means  of  it  to 
get  any  part  of  a  given  sect. 

Hint:  To  bisect  a  sect,  open  the  compasses 
so  that  the  distance  from  10  to  10  is  equal  to  the 
given  sect.  Then  the  distance  from  5  to  5  equals 
one-half  the  given  sect. 

What  part  of  the  given  sect  would  the  distance  from  6  to  6  be? 

c466.  Show  how  by  using  the  sector  compasses  to  divide  a  given 
sect  into  10  equal  parts. 

c467.  The  sector  compasses  may  be  used  to  find  the  fourth 
proportional  to  three  given  sects  as  follows :  From  center  0  on  OL 

mark  off  OA  equal  to  a.     Open  the  a ^ 

sector    until   the   transverse   distance  & 

at  A  equals  6.     Then  if  OB  be  marked 

off  on  OL  equal  to  c,  the  transverse  • 

distance  at  B  is  the  required  fourth  proportional.    Why? 

*  c  is  prefixed  to  the  numbers  of  exercises  better  suited  to  class  dis- 
cussions than  to  written  or  home  work. 
9 


130 


PLANE  GEOMETRY 


Theorem  36.    //  the  ratio  of  the  sides  of  one  triangle  to  those 
of  another  is  constant,  the  triangles  are  similar. 

Given:  A  ABC  and  A  AiB^d  with  -  =  ~  =  -. 

a       b      c 

To  prove:    A  ABC  <° 
A  AiBiCi. 

Suggestions  for  proof: 


AY  =  A^.    DrawjfF. 

Prove  A  ABC  ™  A  AXY. 
Prove  A  AXY 

a       b 
NOTE  :    —  =  —  = 


,  and  on  AC  lay  off 


c       .  .a     b 

-  (given)  and  -  =  - 

Ci  Z       X 


(Why?)    buty=ci. 


G 


E 


EXERCISES.     SET  XLVIII  (concluded) 

d468.  In  the  figure,  E,  F,  G,  and  H  are  the  mid-points  of  the 
sides  of  the  square  ABCD,  and  the  points    j)m 
are  joined  as  shown. 

Show  that  the  following  triangles  are 
similar: 

(a)  EBC,  ELB,  ELY,  and  EUN;  (b) 
BLZ,BST,tmdBXO;  (c)  EYZ  and  EUC; 
(d)  YLZ,  YMB,  and  BHP;  (e)  BOY  and 
BHD;  (/)  BEZ  and  ATB;  (g)  BYZ  and 
BHT;  (h)  BYF  and  BHC. 

d469.  The  following  gives  a  procedure  used  in  surveying  for 
running  a  line  through  a  given  point  parallel  to  a  wholly  inacces- 
B         sible  line.     Study  the  diagram  and 
notes  and  then  justify  the  method. 
NOTES  :    Take  C  in  sect  MB.    Select 
D  at  any  convenient  place.    Run  M  F  \  \  DB. 
Find  E  in  MF  in  line  with  D  and  C.    Run 
EN  ||  AD,  meeting  AC  at  N.    Then  MN 
\\AB. 

Why  is  it  that  we  can  run  paral- 
lels through  M  and  D,  whereas  we  cannot  run  the  one  through  M 
directly? 


SIMILARITY 


131 


d470.  The  pantograph,  invented  in  1603  by  Christopher 
Schemer,  is  an  instrument  for  drawing  a  plane  figure  similar  to  a 
given  plane  figure,  and  is,  hence,  useful  for  enlarging  and  re- 
ducing maps  and  diagrams. 

The  pantograph,  shown  in  two  positions,  consists  of  four  bars  so 
pivoted  at  B  and  E  that  the  opposite  bars  are  parallel.  Pencils  are 
carried  at  D  and  F,  and  A  turns  upon  a  fixed  pivot.  BD  and  DE  may 

A  /?  J?J?  A  D 

be  so  ad  jus  ted  as  to  make  the  ratio  of  -T-~  (and  hence  ^-^and-r-^) 


whatever  is  desired.     So  if  F  traces  a  given  figure,  D  will  trace  a 

AD 
similar  one,  the  ratio  of  similitude  being  the  fixed  ratio  -r-^. 


C 


FIG.  1 


FIG.  2 


(a)  Prove  that  A,  D,  and  F  are  always  in  the  same  straight  line, 
(6)  Prove  that  -j-=  is  constant  and  equal  to  -r-^. 


FIG.  3 


FIG.  4 


(c)  Make  a  pantograph.    A  crude  one  can  be  made  of  stiff  card- 
board and  brass  brads. 

NOTE:  Figs.  3  and  4  show  interesting  elaborations  of  the  pantograph. 

471.  Summarize  the  conditions  necessary  and  sufficient  to  make 
triangles  similar. 

472.  How  may  four  sects  be  proved  proportional? 


132 


PLANE  GEOMETRY 


D.  PERIMETERS  AND  AREAS  OF  SIMILAR  TRIANGLES 

Theorem  37.     The  perimeters  of  similar  triangles  are  propor- 
tional to  any  two  homologous  sides,  or  any 
two  homologous  altitudes. 


Cl       xl 

Given:  Aa&coo  A«i6iCi  with  hA.b  and  /YiJ_6i. 
a+6+c        a 


To  prove: 

Suggestions  for  proof: 
a 


b         c          h 
—  (or  —  or  — )  =  — . 
61       ci         hi 


=  —  =  -.    WThy? 

3+6+c        a 

=  — .     Why? 


h         a 
Prove —  =  —  by  showing  A  CBX  v>  ACiBiXi. 

Cor.  1.    Any  two  homologous  altitudes  of  similar  triangles  have 
the  same  ratio  as  any  two  homologous  sides. 

Theorem  38.    The  areas  of  similar  triangles  compare  as  the 
squares  of  any  two  homologous  sides. 

Given:  Aa6c  <*>  AaifriCj. 

Aa6c        c2    /     62 
To  prove: 


Suggestions  for  proof: 

A<*  *'.i»L.       Why, 

/iiOi       hi  bi 


- 
61 


-.     Why? 


a2    /      62        c2  \ 

^r^orc7j- 


SIMILARITY 


133 


EXERCISES.     SET  XLIX.     AREAS  OF  SIMILAR  TRIANGLES 

473.  Draw  a  triangle.    Construct  a  second  one  similar  to  it, 
having  an  area  nine  times  as  great. 

474.  Connect  the  mid-points  of  two  adjacent  sides  of  a  parallelo- 
gram.    What  part  of  the  area  of  the  whole  figure  is  the  triangle 
thus  formed? 

475.  Fold  a  rectangular  sheet  of  paper  from  one  corner  as  shown. 
The  successive  creases  are  to  be  equally  distant  from  each  other 
and  parallel.    Prove  that  the  ratio  of 

the  successive  areas  between  creases  is 
1  to  3  to  5  to  7,  etc. 

476.  When,    in    forestry,    shadows 
cannot  be  used,  justify  the  following 
method  of  getting  the  height  of  a  tree. 
A   staff  is    planted    upright    in    the 
ground.     A  man  sights  from  S  to  the 
top  and  foot  of  the  tree.     His  assistant 
notes  where  his  line  of  sight  crosses 
the  staff. 

(a)  What  measurements  does  he  need  to  take? 

(6)  Assume  a  reasonable  set  of  data  and  calculate  AB. 

E.  APPLICATIONS  OF  SIMILAR  TRIANGLES 

Theorem  39.  The  altitude  upon  the  hypotenuse  of  a  right 
triangle  divides  the  triangle  into  triangles  similar  each  to  each 

and  to  the  original. 

Given :  A  abc  with  a  _|_  b  and  h  ±c- 
To   prove:  ABCH  ^  ACHA  <» 

ABC. 
Suggestions  for  proof: 

%-B   is  common  to  the  two 
right  triangles  BCH  and  ABC. 
4-A  is  common  to  the  two  right  triangles  CHA  and  ABC. 

I.  PROJECTIONS 

The  word  " projection"  has  a  variety  of  meanings  in  general  use. 
We  refer  to  projecting  ourselves  into  a  situation;  or  to  projecting 
a  picture  on  a  screen  by  means  of  a  lantern.  In  geometry  the  word 


134 


PLANE  GEOMETRY 


has  a  technical  significance  which  we  exemplify  in  the  following 
little  experiment.  Place  in  the  sunlight  a  table  covered  with  a 
white  cloth.  Hold  over  the  table  (parallel  to  it)  a  plate  of  thin 
glass  on  which  small  figures  of  dark  paper  have  been  pasted. 
The  sun  will  project  these  diagrams  on  the  cloth.  Upon 
reflection  you  will  note  a  certain  relationship  between  the  point 
or  sect  represented  on  the  glass  and  its  manifestation  on  the  cloth. 

To  use  scientifically  the  idea  of  projection  we  need  exact  defini- 
tions rather  than  these  vague  assumptions.  Let  us  express  these 
ideas  in  geometrical  terms. 

The  intersection  of  a  perpendicular  to  a  line  with  that  line  is  called 
the  foot  of  the  perpendicular.  The  projection  of  a  point  on  a  line 
is  the  foot  of  the  perpendicular  from  the  point  to  the  line.  The  pro- 
jection of  a  sect  on  a  line  is  the  sect  cut  off  by  the  projections  of  its 
extremities  on  that  line.  For  example: 


*  F  B 

F  is  projection  of  P  on  AB. 


F       B   A       E 


EF  is  projection  of  CD  on  AB. 

a 


E 


CF  is  projection  of  CD  on  AB.  EF  is  the  projection  of  CD  on  AB. 

Would  the  projection  ever  be  as  long  as  the  original  sect?    Ever 
longer? 

NOTE:  These  projections  are  sometimes  referred  to  as  orthogonal  to  dis- 
tinguish them  from  other  types  met  with  in  higher  mathematics. 

The  notion  of  projections  was  originally  obtained  from  that  of 
shadows.  The  projection  of  a  circle  in  one  plane  on  another  plane 
was  its  shadow.  It  is  evident  that  a  scientific  study  of  shadows 
becomes  very  complicated.  Consider,  for  instance,  the  effect  on 
the  shadow  caused  by  the  various  relative  positions  of  the  planes 
and  the  positions  of  the  light.  Projective  geometry,  an  advanced 


SIMILARITY 


135 


study,  concerns  itself  with  the  more  complex  phases  of  the  subject. 
In  elementary  geometry,  we  refer  only  to  one  very  small  instance 
of  shadow  geometry,  reduced,  as  you  note,  to  geometric  definitions. 
These  projections  we  refer  to  as  right  or  orthogonal. 
Cor.  1.    Each  side  of  a  right 

triangle  is  a  mean  propor- 

tional between  the  hypote- 

nuse   and    its    projection 

upon  the  hypotenuse. 
NOTE:  &BCH  «>  ACBA^Fonn  a  B 

proportion  involving  BC,  I3A,  BH. 

Cor.  2.    The  square  of   the  hypotenuse  of  a  right  triangle  is 
equal  to  the  sum  of  the  squares  of  the  other  two  sides. 

NOTE:  a*=c.BH,  and  bz=c-HA.    Why? 


This  fact  is  very  important  in  geometry,  and  has  an  interesting 
history.  The  first  proof  of  the  theorem  is  attributed  to  Pythagoras 
about  500  B.  c.,  although  the  fact  was  known  much  earlier. 
Reference  has  already  been  made  to  the  history  of  the  theorem  in 
Chapter  I.  Its  later  development  consists  of  numerous  proofs 
worked  out  by  later  mathematicians.  In  the  following  exercises 
specimens  of  such  will  be  found,  and  further  interesting  proofs  are 
contained  in  Heath's  Monograph,  "The  Pythagorean  Theorem." 

EXERCISES.     SET  L.    PROJECTIONS.    PYTHAGOREAN  RELATION 

477.  What  would  be  the  projection  of  a  sect  10"  long  on  a  line 
with  which  it  makes  an  angle  of  (a)  30°?    (6)  45°?   (c)  60°? 

478.  In  the  sixteenth  century  the  distance  from  A  to  the  inac- 
cessible point  B  was  determined  by  means  of  an  instrument  called 

the  "squadra."  The  squadra,  like 
a  modern  carpenter's  square,  con- 
sisted of  two  metallic  arms  at  right 
angles  to  each  other.  To  measure 
AB  the  squadra  was  supported,  as 
in  the  figure,  on  a  vertical  staff 
AC.  One  arm  was  pointed  toward 
B,  and  the  point  D  on  the  ground,  at  which  the  other  arm 


136 


PLANE  GEOMETRY 


pointed,  was  noted.     By  measuring  AD  and  AC,  show  how  AB 
may  be  computed. 

d479.  Two  stakes  are  set  on  a  hillside  whose  slope  is  20%  (i.e., 
20  ft.  rise  in  100  ft.  measured  along  the  slope).  The  distance 
between  the  stakes,  measured  along  the  slope,  is  458  ft.  What  is 
the  horizontal  distance  between  them? 

480.  The  accompanying  drawing 
represents  a  plot  of  land  divided  as 
indicated.  DE  =  22'8",  EB  =  1W, 
BF  =  50',  FG  (alley)  =  16',  GH  =  150', 
HI  =  66',  and  IK  =  232'6".  Find  the 
length  of  AB  in  feet,  and  the  area  of 
triangle  ACD  in  square  rods. 
d481.  The  figure  shows  a  ground  plan  of  a  zigzag  or  "worm" 
fence.  The  rails  are  11  feet  long,  and  a  lap  of  1  foot  is  allowed 
at  each  corner.  Stakes,  supporting 
the  rider  rails,  are  set  along  the 
boundary  line.  Find  the  amount 
of  ground  wasted  by  the  construc- 
tion of  such  a  fence  100  rods  long. 

How  much  more  fence  is  needed  in  this  zigzag  fence  than  in  a 
straight  one? 

d482.  Let  ABC  be  any  right-angled  triangle,  right-angled  at  C, 
and  let  the  square  ABDE  be  described  on  the  hypotenuse  AB, 
overlapping  the  triangle.  Prove  that  the  perpendicular  from  E 
upon  AC  is  of  length  b,  and  hence  that  the  area  of  the  triangle 
ACE  is  ^62.  Similarly,  prove  that  the  area  of  the  triangle  BCD 
is  %a2.  Notice  that  these  two  triangles  have  equal  bases  c  and 
total  height  c.  Hence  prove  that  a2-j-62  =  c2. 

483.  The  great  Hindu  mathematician, 
Bhaskara  (born  1114  A.  D.),  proceeds  in  a 
somewhat  similar  manner.  He  draws  this 
figure,  but  gives  no  proof.  It  is  evident  that 
he  had  in  mind  this  relation: 


Boundary 


Line 


Give  a  proof. 


SIMILARITY 


137 


d484.  A  somewhat  similar  proof  can  be  based  upon  the  following 
figure : 

If  the  four  triangles,  1  +  2+3+4,  are  taken 
away,  there  remains  the  square  on  the  hypo- 
tenuse. But  if  we  take  away  the  two  shaded 
rectangles,  which  equal  the  four  triangles,  there 
remain  the  squares  on  the  two  sides.  Therefore 
the  square  on  the  hypotenuse  equals  the  sum  of  these  two  squares. 
Give  details  of  the  proof. 

d485.  This  exercise  makes  the  Pythagorean  Theorem  a  special 
case  of  a  proposition  due  to  Pappus  (fourth  century  A.  D.),  relating 
to  any  kind  of  triangle. 

Somewhat  simplified,  this  proposition  asserts  that  if  ABC  is 
any  kind  of  triangle,  and  MC,  NC  are  parallelograms  on  AC,  BC, 

the  opposite  sides  being  produced 
to  meet  at  P;  and  if  PC  is  produced 
making  QR  =  PC;  and  if  the  paral- 
lelogram AT  is  constructed,  then 
AT=MC-\-NC. 

For  MC=AP?=AR,  having  equal 
altitudes  and  bases. 
Similarly,  NC^QT.      Adding,  MC+NC=AT. 
If,  now,  ABC  is  a  right  triangle,  and  if  MC  and  NC  are  squares, 
it  is  easy  to  show  that  A  T  is  a  square,  and  the  proposition  reduces 
to  the  Phythagorean  Theorem.    Show  this. 

d486.  The  Arab  writer,  Al-Nairizi  (died  about  922  A.D.),  attri- 
buted to  Thabit  ben  Korra  (826-901  A.D.)  a  proof  substantially 
as  follows: 

The  four  triangles  T7  can  be  proved  congruent. 
Then  if  we  take  from  the  whole  figure  T  and 
Ti,  we  have  left  the  squares  on  the  two  sides 
of  the  right  angle.  If  we  take  away  the  other 
two  triangles  instead,  we  have  left  the  square 
on  the  hypotenuse.  Therefore  the  former  is 
equivalent  to  the  latter.  Give  details  of  proof. 

d487.  A  proof  attributed  to  the  great  artist,  Leonardo  da  Vinci 
(1452-1519),  is  as  follows: 


138 


PLANE  GEOMETRY 


Q^ 


The  construction  of  the  following  figure  is  evident.   It  is  easily 

shown  that  the  four  quadrilaterals 
ABMX,  XNCA,  SBCP,  and  SRQP 
are  congruent. 

/.  ABMXNCA  equslaSBCPQRS 
but  is  not   congruent   to   it,  the 
*M  congruent  quadrilaterals  being  dif- 
ferently arranged. 

Subtract  the  congruent  triangles 
MXN,  ABC,  RAQ,  and  the  pro- 
position is  proved.  a  b  A 


Give  details  of  proof. 

d488.  A  proof  attributed  to  President  Garfield  is 
suggested  by  the  accompanying  diagram.  Work  itout.  a 

NOTE:  a=ait  b=bi,  c=ci.  ABDC  is  a  trapezoid.  What  is 
the  altitude  of  the  trapezoid?  Its  bases?  Its  area?  How 
else  may  the  area  of  the  trapezoid  be  found? 

d489.  Show  that  if  AB=a  (in  Fig.  3), 

(a)  #C^ 


(6)  BLm 


H 


E 


490.  In  the  middle  of  a  pond   10  ft. 
square  grew  a  reed.    The  reed  projected 
1  ft.   above  the   surface  of   the   water. 
When  blown  aside  by  the  wind,  its  top 
part  reached  to  the  mid-point  of  a  side 
of  the  pond.     How  deep  was  the  pond? 
(Old  Chinese  problem.) 

491.  Show    that    the    following    dia- 
grams illustrate  methods  of  representing 


FIG.  3. 

Moorish  Design,  from  Mabel    the  square  roots  of  integers. 


Sykes'  Source  Book  of  Problems 
for  Geometry. 


B 


=  BE  =  \3,  etc. 


C     D    E  F  etc. 


SIMILARITY 


139 


AC  is  a,  square. 


D 


E 


AG=AE=\/3,  etc. 

NOTE  :  Such  methods  would  not  be 
practical  for  large  numbers     Why?      J5 


H 


The  first  of  these  methods  is  used  on  the  "line  of  squares"  on 
the  sector  compasses. 

492.  The  Hindus  said  that  triangles  having  the  following  sides 
are  right  triangles.  How  is  the  assumption  they  apparently  made 
related  to  Theorem  40,  Cor.  2?  (a)  5,  12,  13.  (6)  15,  36,  39. 
(c)  8,  15,  17.  (d)  12,  35,  37. 

II.  TRIGONOMETRIC  RATIOS 

Let  ABC  be  an  acute  angle.  Drop  a 
series  of  perpendiculars  to  BA  from 
any  points  on  BC. 

It  will  readily  be  noted  that  the  right 
triangles  formed,  BPiRi,  BP^Rz,  etc.. 


A  are  similar,  having  the  angle  B  in  com- 
mon.  The  equality  of  the  following  ratios  will  result : 


(D 


-BPi 


BP^ 


BP,    BP2 


^5  =  ^  and  (3) 


Why? 


BP '4     BP&  BRi      JoRz      BR$      BR±      BR$ 

If  we  think  of  $iABC  as  being  generated  by  sect  BC  revolving 
counterclockwise  from  the  position  BA  to  BC  we  may  call  BA  the 
initial  side  and  BC  the  terminal  side.  These  perpendiculars  from 
points  on  the  terminal  side  may  be  thought  of  as  projectors  form- 
ing on  the  initial  side  the  projections  of  sects  of  the  terminal  side. 
We  may  then  summarize  the  facts  given  as  ratios  in  what  preceded 
as  follows :  For  any  acute  angle  if  perpendiculars  be  dropped  to  the 
initial  side  from  any  points  on  the  terminal  side  the  ratios  (1)  of 
the  projector  to  the  sect  of  the  terminal  side,  (2)  of  the  projection  to  the 
sect  of  the  terminal  side,  and  (3)  of  the  projector  to  the  projection  of  a 
sect  of  the  terminal  upon  the  initial  side  are  constants.  These  ratios 
are  given  the  names  sine,  cosine,  and  tangent,  respectively. 


140 


PLANE  GEOMETRY 


Initial  Side 


TP 

Thus  (1)  the  sine  of  angle  ABC  is=- 

BP 

(2)  The  cosine  of  angle  ABC  is  = 

~jp 

-^    (3)  The  tangent  of  angle  ABC  is  — 

BJ 

These  are  referred   to  as  trigonometric  ratios   of  an  angle. 

If  in  the  fixed  A  ABC,  right  angled 
at  C,  the  side  opposite  A  is  called  o, 
the  side  adjacent,  a,  and  the  hypote- 
nuse, h,  fill  in  the  following: 

EXERCISES.     SET  LI.     TRIGONOMETRIC  RATIOS 

493.  Make  a  table  of  the  values  of  the 
sines,  cosines,  and  tangents  of  angles  of  30°, 
45°,  and  60°. 

494.  Showbymeans  of similiar  trianglesthat 
the  sine  of  60°  is  the  same  as  the  cosine  of  30°. 

d495.  When  a  wagon  stands  upon  an  incline,   its  weight  is 
resolved  into  two  forces,  one  the  pressure  against  the  incline,  the 

other  tending  to  make  it  run  down  the 
incline.     Show  that  the  force  along  the 
incline  is  to  the  weight  of  the  wagon  as 
the  height  of  the  incline  is  to  its  length. 
If  the  incline  makes  a  30°  angle  with  the 
horizontal,  with  what  force  does  a  loaded 
wagon,  weighing  three  tons,  tend  to  run  down  the  incline,  i.e.,  disre- 
garding friction,  what  force  must  a  team  exert  to  pull  it  up  the  slope? 
496.  Fill  out  the  following  table: 


Polygon 

Dimensions 

Perimeter 

Area 

Parallelogram 

Base  18 
Angle  60° 

300 

300 

Rectangle 

Base  18 

300 

300 

Rhombus 

Angle  60° 

300 

300 

Square 

300 

300 

SIMILARITY 


141 


D 


497.  To  measure  the  height  of  an  object  AB  by  drawing  to  scale: 
Measure  a  distance  CD  towards  A.    Measure  the  angle  ACB 

and  the  angle  ADB.    Then  draw  a  plan  thus:   Representing,  say, 

100  ft.  by  a  sect  an  inch  long,  draw 

EF  to  represent  CD  in  the  plan, 

and  draw  the  angle  HFG  equal  to 

the  angle    ADB,   and    the    angle 

PEG  equal  to  the  angle  ACB,  and 

draw  GH  at  right   angles  to  EF 

prolonged.   Measure  GH. 

Show  how  to  compute  A  B. 

(a)  If  CD  =  175  ft.,  the 
angle   ADB  =  45°,   and 

angle  DCB  =  30°,  compute      </      E 
AB. 

(b)  Draw  a  diagram  to  scale  and  compare  the  result  with  that 
obtained  from  your  calculations. 

(c)  Which  of  these  results  is  more  accurate?    Why? 

498.  If  a  survey  is  made,  using  a  100  ft.  tape,  and  on  a  hill, 
the  lower  chainman  holds  his  end  of  the  tape  2  ft.  too  low: 

(a)  What  error  will  be  caused  in  one  tape  length? 
(6)  If  the  distance  between  two  stations  on  the  hillside  is 
recorded  as  862  ft.,  what  is  the  actual  distance? 

(c)  If  the  problem  is  to  lay  off  adistanceof 
900  ft.,  what  is  the  actual  distance  laid  off? 

499.  A  ladder  30  ft.  long  leans  against 
the  side  of  a  building,  its  foot  being  15  ft. 
from  the  building.     What  angle  does  the 
ladder  make  with  the  ground? 

500.  In    order    to 
find  the  width  of  a 
river,  a  distance  AB 
was  measured  along 

the  bank,  the  point  A  being  directly  opposite 
a  tree  C  on  the  other  side.  If  the  angle  ABC 
was  observed  to  be  60°,  and  AB  100  ft.,  find 
the  width  of  the  river. 


15 


B A 


142 


PLANE  GEOMETRY 


501.  A  house  30  ft.  wide  has  a  gable  roof  whose  rafters  are  20 
ft.  long.    What  is  the  pitch  of  the  roof?    (The  pitch  is  the  angle 
between  a  rafter  and  the  horizontal.) 

502.  A  barn  60  ft.  wide  has  a  gable  roof  whose  rafters  are  30\/2 
ft.  long.    What  is  the  pitch  of  the  roof,  and  how  far  above  the 

eaves  is  the  ridgepole? 

The  angle  of  elevation  is  the  angle  between 
the  ray  of  light  from  the  object  to  the  eye  and 
the  horizontal  line  in  the  same  plane,  when  the 
object  is  above  the  horizontal  line.  When  the 
object  observed  is  below  the  horizontal  line,  the 
angle  is  called  the  angle  of  depression. 


For  example: 


Eye 


Horizontal 


Object 


Eye 


Horizontal 


Object 


EXERCISES.     SET  LI  (continued) 

503.  At  a  point  200  ft.  in  a  horizontal  line  from  the  foot  of  a 
tower  the  angle  of  elevation  of  the  top  of  the  tower  is  observed  to 
be  60°.    Find  the  height  of  the  tower. 

504.  The  vertical  central  pole  of  a  circular  tent  is  20  ft.  high, 
and  its  top  is  fastened  by  ropes  40  ft.  long  to  stakes  set  in  the 
ground.    How  far  are  the  stakes  from  the  foot  of  the  pole,  and 
what  is  the  inclination  of  the  ropes  to  the  ground? 

505.  At  a  point  midway  between  two  towers  on  a  horizontal 
plane  the  angles  of  elevation  of  their  tops  are  30°  and  60°  respec- 
tively.   Show  that  one  tower  is  three  times  as  high  as  the  other. 

506.  A  flagstaff  25  ft.  high  stands  on  the  top  of  a  house.    From 
a  point  on  the  plane  on  which  the  house  stands,  the  angles  of 
elevation  of  the  top  and  the  bottom  of  the  flagstaff  are  observed 
to  be  60°  and  45°  respectively.    Find  the  height  of  the  house. 

507.  A  man  walking  on  a  straight  road  observes  at  one  mile- 
stone a  house  in  a  direction  making  an  angle  of  30°  with  the  road, 
and  at  the  next  milestone  the  angle  is  60°.    How  far  is  the  house 
from  the  road? 


SIMILARITY 


143 


508.   Find  the  number  of  square  feet  of  pavement  required  for 
the  shaded   portion   of  the   streets 
shown  in  the  figure,  all  the  streets 
being  50  ft.  wide. 

It  is  not  possible  to  determine  the  g[ 

trigonometric  ratios  of  angles  other  gtfVI  w C 
than  30°,  45°,  and  60°  by  elementary 
plane  geometry.  By  the  use  of  the  protractor  any  acute  angle  can 
be  drawn,  and  with  a  ruled  edge  the  sects  needed  may  be  meas- 
ured and  approximations  may  be  made  for  the  ratios.  The  values 
correct  to  many  decimal  places  have  been  scientifically  worked 
out  and  tabulated.  A  table  correct  to  four  places  follows  for 
use  in  subsequent  problems.  Corrections  for  fractions  of  min- 
utes may  be  made  as  in  the  case  of  logarithmic  tables. 


Deg- 

Sine 

Cosine 

Tangent 

Deg. 

Sine 

Cosine 

Tangent 

Deg. 

Sine 

Cosine 

Tangent 

1 

.0175 

.9998 

.0175 

31 

.5150 

.8572 

.6009 

61 

.8746 

.4848 

1.8040 

2 

.0349 

.9994 

.0349 

32 

.5299 

.8480 

.6249 

62 

.8829 

.4695 

1.8807 

3 

.0523 

.9986 

.0524 

33 

.5446 

.8387 

.6494 

63 

.8910 

.4540 

1.9626 

4 

.0698 

.9976 

.0699 

34 

.5592 

.8290 

.6745 

64 

.8988 

.4384 

2.0503 

5 

.0872 

.9962 

.0875 

35 

.5736 

.8192 

.7002 

65 

.9063 

.4226 

2.1445 

6 

.1045 

.9945 

.1051 

36 

.5878 

.8090 

.7265 

66 

.9135 

.4067 

2-2460 

7 

.1219 

.9925 

.1228 

37 

.6018 

.7986 

.7536 

67 

.9205 

.3907 

2.3559 

8 

.1392 

.9903 

.1405 

38 

.6157 

.7880 

.7813 

68 

.9272 

.3746 

2.4751 

9 

.1564 

.9877 

.1584 

39 

.6293 

.7771 

.8098 

69 

.9336 

.3584 

2.6051 

10 

.1736 

.9848 

.1763 

40 

.6428 

.7660 

.8391 

70 

.9397 

.3420 

2.7475 

11 

.1908 

.9816 

.1944 

41 

.6561 

.7547 

.8693 

71 

.9455 

.3256 

2.9042 

12 

.2079 

.9781 

.2126 

42 

.6691 

.7431 

.9004 

72 

.9511 

.3090 

3.0777 

13 

.2250 

.9744 

.2309 

43 

.6820 

.7314 

.9325 

73 

.9563 

.2924 

3.2709 

14 

.2419 

.9703 

.2493 

44 

.6947 

.7193 

.9657 

74 

.9613 

.2756 

3.4874 

15 

.2588 

.9659 

.2679 

45 

.7071 

.7071 

1.0000 

75 

.9659 

.2588 

3.7321 

16 

.2756 

.9613 

.2867 

46 

.7193 

.6947 

.0355 

76 

.9703 

.2419 

4.0108 

17 

.2924 

.9563 

.3057 

47 

.7314 

.6820 

.0724 

77 

.9744 

.2250 

4.3315 

18 

.3090 

.9511 

.3249 

48 

.7431 

.6691 

.1106 

78 

.9781 

.2079 

4.7046 

19 

.3256 

.9455 

.3443 

49 

.7547 

.6561 

.1504 

79 

.9816 

.1908 

5.1446 

20 

.3420 

.9397 

.3640 

50 

.7660 

.6428 

.1918 

80 

.9848 

.1736 

5.6713 

21 

.3584 

.9336 

.3839 

51 

.7771 

.6293 

1.2349 

81 

.9877 

.1564 

6.3138 

22 

.3746 

.9272 

.4040 

52 

.7880 

.6157 

1.2799 

82 

.9903 

.1392 

7.1154 

23 

.3907 

.9205 

.4245 

53 

.7986 

.6018 

1.3270 

83 

.9925 

.1219 

8.1443 

24 

.4067 

.9135 

.4452 

54 

.8090 

.5878 

1.3764 

84 

.9945 

.1045 

9.5144 

25 

.4226 

.9063 

.4663 

55 

.8192 

.5736 

1.4281 

85 

.9962 

.0872 

11.4301 

26 

.4384 

.8988 

.4877 

56 

.8290 

.5592 

1.4826 

86 

.9976 

.0698 

14.3006 

27 

.4540 

.8910 

.5095 

57 

.8387 

.5446 

1.5399 

87 

.9986 

.0523 

19.0811 

28 

.4695 

.8829 

.5317 

58 

.8480 

.5299 

1.6003 

88 

.9994 

.0349 

28.6363 

29 

.4848 

.8746 

.5543 

59 

.8572 

.5150 

1.6643 

89 

.9998 

.0175 

57.2900 

30 

.5000 

.8660 

.5774 

60 

.8660 

.5000 

1.7321 

90 

1.0000 

.0000 

00 

144  PLANE  GEOMETRY 

In  trigonometry  a  more  extended  study  of  these  ratios  will  be 
given.  The  ratios  of  angles  of  more  than  90°  will  be  considered, 
and  other  ratios  which  are  constant  will  be  developed. 

Work  involving  calculations  with  the  trigonometric  ratios  is 
often  simplified  by  the  use  of  tables  of  logarithmic  functions.  For 
this  purpose,  and  for  greater  facility  in  the  use  of  logarithms  in 
general,  and  in  the  use  of  the  natural  functions,  it  would  be  to  the 
pupil's  advantage  to  procure  a  compact  volume  of  tables.  An 
excellent  book  for  this  purpose,  costing  only  twenty  cents,  is 
Prof.  A.  Adler,  Fiinfstellige  Logarithmen  (Sammlung  Goschen). 

EXERCISES.     SET  LI  (concluded) 

509.  The  sect  AB  15  inches  long  makes  an  angle  of  35°  with  the 
line  OX.    Find  its  projection  on  OX.    Find  its  projection  on  the 
line  0  Y  perpendicular  to  OX  and  in  the  same  plane  as  OX  and  AB. 

510.  What  is  the  angle  of  the  sun's  altitude  if  the  shadow  of  a 
telegraph  pole  30  ft.  high  is  40  ft.  long? 

511.  A  tower  is  6 15  ft.  high.    How  large  an  angle  does  it  subtend 
at  a  point  which  is  1^  mi.  away  and  on  the  same  horizontal  plane 
as  its  base? 

512.  A  mariner  finds  that  the  angle  of  elevation  of  the  top  of  a 
cliff  is  16°.    He  knows  from  the  location  of  a  buoy  that  his  distance 
from  the  foot  of  the  cliff  is  half  a  mile.    How  high  is  the  cliff? 

513.  At  40  ft.  from  the  base  of  a  fir  tree  the  angle  of  elevation 
of  the  top  is  75°.    Find  the  height  of  the  tree. 

514.  A  flagstaff  75  ft.  high  casts  a  shadow  40  ft.  long.    Find  the 
angle  of  elevation  of  the  sun  above  the  horizon. 

515.  To  find  the  distance  across  a  lake  from  a  point  A  to  a 
n  point  B}  a  man  measured  100  rods  to  a  point 

~  C  on  a  line  perpendicular  to  the  line  AB, 

and  found  that  the  angle  ABC  was  50°. 
How  could  he  find  the  distance  across  the 
lake?  What  is  the  distance? 

516.  What  is  the  angle  of  slope  of  a  road 
bed  that  has  a  grade  of  5  per  cent?  One  with 
a  grade  of  25  hundredths  per  cent?  (By  "a  grade  of  5  per  cent" 
is  meant  a  rise  of  five  feet  in  a  horizontal  distance  of  one  hun- 


SIMILARITY 


145 


dred  feet.    By  "the  angle  of  the  slope"  of  such  a  grade  is  meant 
the  angle  whose  tangent  is  0.05.) 

517.  A  steamer  is  moving  in  a  southeasterly  direction  at  the 
rate  of  25  miles  an  hour.    How  fast  is  it  moving  in  an  easterly 
direction?    In  a  southerly  direction? 

518.  A  balloon  of  diameter  50  ft.  is  directly  above  an  observer 
and  subtends  a  visual  angle  of  4°.    What  is  the  height  of  the 
balloon? 

d519.  The  angle  of  elevation  of  a  balloon  from  a  point  due  south 
of  it  is  60°,  and  from  another  point  1  mile  due  west  of  the  former, 
the  angle  of  elevation  is  45°.  Find  the  height  of  the  balloon. 

520.  Wishing  to  determine  the  width  of  a  river,  I  observed  a 
tree  standing  directly  across  the  bank.     The  angle  of  elevation 
of  the  top  of  the  tree  was  32°;  at  150  ft.  back  from  this  point,  and 
in  the  same  direction  from  the  tree,  the  angle  of  elevation  of  the 
top  of  the  tree  was  21°.    Find  the  width  of  the  river. 

521.  A  tree  is  standing  on  a  bluff  on  the  opposite  side  of  the 
river  from  the  observer.    Its  foot  is  at  an  elevation  of  45°,  and  its 
top  at  60°.    (a)  Compare  the  height  of  the  bluff  with  that  of  the 
tree  (i.e.,  find  the  ratio).    (6)  What  measurement  would  you  use 
to  find  the  height  of  the  tree?    (c)  The  height  of  the  bluff?    (d) 
The  width  of  the  river? 

d522.  Two  men  are  lifting  a  stone  by 
means  of  ropes.  As  the  stone  leaves  the 
ground  one  man  is  pulling  with  a  force  of  / 

85  Ibs.  in  a  direction  25°  from  the  vertical,  / 

while  the  other  man  is  pulling  at  an  angle  / 

of  40°  from  the  vertical.  Determine  the         R  / 
weight  of  the  stone.  \ 

523.  A  50  ft.  pole  stands  on  the  top  of  <£*V 

a  mound.    The  angles  of  elevation  of  the  ___ 

top  and  the   bottom    of    the    pole    are  S 

respectively  35°  and  62°.    Find  the  height  of  the  mound. 

524.  From  the  top  of  a  mountain  1050  ft.  high  two  buildings 
are  seen  on  a  level  plane,  and  in  a  direct  line  from  the  foot  of  the 
mountain.    The  angle  of  depression  of  the  first  is  35°,  and  of  the 
second  is  21°.    Find  the  distance  between  the  two  buildings. 

10 


146 


PLANE  GEOMETRY 


525.  Certain  lots  in  a  city 


B      Street 

shown  in  the  figure. 
Find  the  distance  on 
a  straight  line  from 
A  to  E.  (Use  log- 
arithmic tables.) 

527.     With    data, 
find  the  length  and  bearing 
logarithmic  tables.) 


are  laid  out  by  lines  perpendicular 
to  B  Street  and  running 
through  to  A  Street,  as 
shown  in  the  figure.  Find 
the  widths  of  the  lots  on  A 
Street  if  the  angle  between 
the  streets  is  28°  40'. 

526.  In  surveying 
around  an  obstacle  meas- 
urements were  taken  as 


of  DF,  a  proposed   street.     (Use 

D 


144.52 


N3.15 


.N  28-30-55.2  E 


d528.  Look  up  and  explain  the  principle  of  the  Vernier.  (Lock 
and  Child,  Trigonometry  for  Beginners  (Macmillan),  is  a  good 
reference  for  this  point — pp.  120-126). 

529.  The  shadow  of  a  vertical  10  ft.  pole  is  14  ft.  long.    What 
is  the  angle  of  elevation  of  the  sun? 

530.  The  tread  of  a  step  on  a  certain  stairway  is  10"  wide;  the 
step  rises  1"  above  the  next  lower  step.    Find  the  angle  at  which 
the  stairway  rises. 

531.  The  width  of  the  gable  of  a  house  is  34  ft.    The  height  of 
the  house  above  the  eaves  is  15  ft.    Find  the  length  of  the  rafters 
and  the  angle  of  inclination  of  the  roof. 

532.  Find  the  angle  between  the  rafter  and  horizontal  in  the 
following  pitch  of  roof:  two-thirds,  one-half,  one-third,  one-fourth. 

533.  Two  trees  M  and  N  are  on  opposite  sides  of  a  river.    A  line 
NP  at  right  angles  to  MN  is  432.7  ft.  long,  and  the  angle  NPM  is 
52°  20'.  What  is  the  distance  fromM  to  JV?  (Use  logarithmic  tables.) 


SIMILARITY  147 

534.  In  an  isosceles  triangle  one  of  the  base  angles  is  48°  20',  and 
the  base  is  18".    Find  the  legs,  the  vertical  angle,  and  the  altitude 
drawn  to  the  base. 

535.  To  find  the  height  of  a  tower,  a  distance  of  311.2  ft.  was 
measured  from  the  foot  of  the  tower,  and  the  angle  of  elevation  of 
the  tower  was  found  to  be  40°  57'.    Find  the  height  of  the  tower. 
(Use  logarithmic  tables.) 

536.  Find  the  shorter  altitude  and  the  area  of  a  parallelogram 
whose  sides  are  10'  and  25'  when  the  angle  between  the  sides  is  74°  33'. 

d537.  The  angle  of  elevation  of  the  top  of  a  spire  from  the  third 
floor  of  a  building  was  35°  12'.  The  angle  of  elevation  from  a 
point  directly  above,  on  the  fifth  floor  of  the  same  building,  was 
25°  33'.  What  is  the  height  of  the  tower  and  its  horizontal  distance 
from  the  place  of  observation,  if  the  distance  between  consecutive 
floors  is  12  ft.,  and  the  first  floor  rests  on  a  basement  5  ft.  above 
the  level  of  the  street? 

538.  (a)  What  size  target  at  33'  from  the  eye  subtends  the  same 
angle  as  a  target  3'  in  diameter  at  987  yds.? 

(6)  Find  the  angle  it  subtends. 

539.  The  summit  of  a  mountain,  known  to  be  14,450  feet  high, 
is  seen  at  an  angle  of  elevation  of  29°  15'  from  a  camp  located  at 
an  altitude  of  6935  feet.    Compute  the  air-line  distance  from  the 
camp  to  the  summit  of  the  mountain.     (Use  logarithmic  tables.) 

d540.  Two  towns,  A  and  B,  of  which  B  is  25  miles  northeast  of 
A,  are  to  be  connected  by  a  new  road.  Ten  miles  of  the  road  is 
constructed  from  A  in  the  direction  N.  23°  E.  What  must  be  the 
length  and  direction  of  the  remainder  of  the  road,  assuming  that  it 
follows  a  straight  line? 

541.  A  car  track  runs  from  A  to  B,  a  horizontal  distance  of  1275' 
at  an  incline  of  7°  45',  and  then  from  B  to  C,  a  distance  of  1585'. 
C  is  known  to  be  509'  above  A.  What  is  the  average  inclination  of 
the  track  from  B  to  C?  (Use  logarithmic  tables.) 

d542.  On  a  map  on  which  1"  represents  1000',  contour  lines 
are  drawn  for  differences  of  100'  in  altitude.  What  is  the  actual 
inclination  of  the  surface  represented  by  that  portion  of  the  map 
at  which  the  contour  lines  are  ^"  apart? 

d543.  The  description  in  a  deed  runs  as  follows :  Beginning  at  a 
stone  (A),  at  the  N.  W.  corner  of  lot  401;  thence  east  112'  to  a 


148  PLANE  GEOMETRY 

stone  (B) ;  thence  S.  36.5°  W.  100';  thence  west  parallel  with  AB  to 
the  west  line  of  said  lot  401 ;  thence  north  on  the  west  line  of  said 
lot  to  the  place  of  beginning.  Find  the  area  of  the  land  described. 

LIST  OF  WORDS  DEFINED  IN  CHAPTER  VI 

Similar  systems  (or  sets)  of  points,  center  of  similitude,  ratio  of  similitude, 
similar  figures.  Projection  of  a  point,  and  of  a  sect,  on  a  line;  projector.  Initial 
and  terminal  side  of  an  angle.  Trigonometric  ratios;  sine,  cosine,  tangent  of 
an  angle.  Angle  of  elevation,  angle  of  depression. 

SUMMARY  OF  THEOREMS  PROVED  IN  CHAPTER  VI 

31.  A  line  parallel  to  one  side  of  a  triangle,  and  cutting  the  other  sides, 
divides  them  proportionally. 

Cor.  1.  One  side  of  a  triangle  is  to  either  of  the  sects  cut  off  by  a  line 
parallel  to  a  second  side,  as  the  third  side  is  to  its  homo- 
logous sect. 

Cor.  2.  A  series  of  parallels  cuts  off  proportional  sects  on  all  trans- 
versals. 

Cor.  3.  Parallels  which  intercept  equal  sects  on  one  transversal, 
do  so  on  all  transversals. 

Cor.  4.  A  line  which  bisects  one-  side  of  a  triangle,  and  is  parallel  to 
the  second,  bisects  the  third. 

32.  A  line  dividing  two  sides  of  a  triangle  proportionally  is  parallel  to  the 
third  side. 

Cor.  1.  A  line  dividing  two  sides  of  a  triangle  so  that  those  sides 
bear  the  same  ratio  to  a  pair  of  homologous  sects  is  paral- 
lel to  the  third  side. 

33.  The  homologous  sides  of  similar  triangles  have  a  constant  ratio,  and 
their  homologous  angles  are  equal. 

34.  Triangles  are  similar  when  two  angles  of  one  are  equal  each  to  each 
to  two  angles  of  another. 

35.  Triangles  which  have  two  sides  of  one  proportional  to  two  sides  of 
another  and  the  included  angles  equal  are  similar. 

36.  If  the  ratio  of  the  sides  of  one  triangle  to  those  of  another  is  constant, 
the  triangles  are  similar. 

37.  The  perimeters  of  similar  triangles  are  proportional  to  any  two  homolo- 
gous sides,  or  any  two  homologous  altitudes. 

Cor.  1.  Homologous  altitudes  of  similar  triangles  have  the  same  ratio 
as  homologous  sides. 

38.  The  areas  of  similar  triangles  compare  as  the  squares  of  any  two  homol- 
ogous sides. 

39.  The  altitude  on  the  hypotenuse  of  a  right  triangle  divides  the  triangle 
into  triangles  similar  to  each  other  and  to  the  original. 

Cor.  1.  Each  side  of  a  right  triangle  is  a  mean  proportional  between 
the  hypotenuse  and  its  projection  upon  the  hypotenuse. 

Cor.  2,  The  square  of  the  hypotenuse  of  a  right  triangle  is  equal  to 
the  sum  of  the  squares  of  the  other  two  sides. 


CHAPTER  VII 

THE  LOCUS 

A.  REVIEW  OF  THE  IDEA  OF  LOCUS  AS  MET  WITH  IN 

ALGEBRA 

I.  REVIEW  AND    SUMMARY   OF    ESSENTIAL  POINTS   IN   THE 
INTRODUCTION  TO  GRAPHIC  MATHEMATICS 

a.  Location  of  Points. 

In  locating  places  on  a  map  we  are  accustomed  to  noting  their 
longitude  and  latitude,  which  means  that  we  refer  to  their  distances 
north  or  south  of  the  equator,  and  east  or  west  of  some  meridian. 
So  we  may  locate  points  on  a  piece  of  paper  by  stating  their  dis- 
tance up  or  down  from  some  fixed  line  of  reference,  and  to  the  right 
or  left  of  some  other  line  of  reference  at  right  angles  to  the  first. 

These  lines  of  reference  are  called  the  axes,  the  distances  up  or 
down  are  called  the  ordinates,  and  those  to  the  right  and  left  the 
abscissas  of  the  points.  The  ordinate  and  the  abscissa  of  a  point 
are  together  called  its  coordinates.  Paper  ruled  off  in  squares  is 
used  for  convenience  in  counting,  and  in  locating  points.  Such 
paper  is  called  coordinate  paper. 

The  abscissa  of  a  point  is  given  first,  followed  by  the  ordinate. 
Plus  or  minus  are  used  in  the  case  of  the  abscissa  to  denote  distance 
to  the  right  or  left  of  the  so-called  y-axis ;  plus  or  minus,  in  the 
case  of  the  ordinate,  denote  distance  above  or  below  the  so-called 
x-axis.  The  intersection  of  the  axes  is  called  their  origin. 

The  coordinates  of  a  point  are  written  in  a  parenthesis  with  a 
comma  between  them;  e.g.  (5,-2)  refers  to  a  point  5  units  to  the 
right  of  the  y-axis  and  2  units  below  the  x-axis. 

EXERCISES.    SET  L1I.    LOCATION  OF  POINTS 

544.  With  reference  to  a  single  pair  of  axes,  plot  the  following 
points  on  a  sheet  of  coordinate  paper: 

(4,5),  (-2,5),  (-2, -5),  (5, -2). 

545.  On  the  same  sheet  plot  also  the  points:  (3,  |),  (  —  3,  — 3), 
(6, -|),  (-3,  -*)• 

546.  Locate  the  points:  (2,  0),  (-5,  0),  (0,  5),  (0,  -|),  (0,  0). 

149 


ISO  PLANE  GEOMETRY 

547.  (a)  All  the  points  on  the  x-axis  have  what  ordinate? 
(b)  All  the  points  on  the  y-axis  have  what  abscissa? 

548.  Construct  the  triangle  whose  vertices  are  (1, 1),  (2,  -2),  (3, 2) . 

549.  Construct  the  quadrilateral  whose   vertices  are  (2,  -1), 
-4, -3),  (-3,5),  (3,4). 

550.  Construct  the  rectangle  whose  vertices  are  (  —  3,  4),  (4,  4), 
-3,  -2),  (4,  -2),  and  find  its  area. 

551.  Construct   the   triangle  whose    vertices    are   (  —  3,   -4), 
(-1,  3),  (2,  -4),  and  find  its  area. 

b.  The  Graph  and  its  Applications. 

The  line  connecting  a  series  of  points  plotted  as  explained  is  called  a 
graph.  Graphs  are  useful  for  giving  information  quickly,  in  making 
estimates,  and  in  the  solution  of  many  problems  such  as  those  involv- 
ing time  and  distance.  We  have  all  seen  the  charts  of  trained 
nurses,  and  newspaper  and  magazine  reports  given  in  graphic  form. 

Among  the  numerous  applications  of  the  graph,  then,  we  may 
list  (1)  records  of  statistics,  (2)  ready  reckoners  which  furnish 
bases  of  interpolation  or  give  convenient  diagrams,  (3)  repre- 
sentations of  formulas  which  make  quick  approximations  possible. 

EXERCISES.     SET  LIII.     APPLIED  PROBLEMS  IN  GRAPHIC 
MATHEMATICS 

552.  Observe  the  readings  of  the  same  thermometer  at  the  same 
hours  daily  for  a  week,  and  record  the  results  of  your  observations 
graphically. 

553.  A  boy  who  can  throw  a  stone  from  a  sling  shot  with  a 
velocity  of  80  ft.  per  second  is  experimenting.    He  finds  that 
when  he  throws  it  in  a  direction  making  an  angle  of  16°  with  the 
ground  it  pitches  35  yds.  away.    This  and  other  results  are  given 
in  the  table  below: 

Angle  (in  degrees) 16    24    32    40    48    56    64     72    80 

Distance  (in  yards) 35     50    60    65     66    61     52     39     22 

(a)  Draw  a  graph  to  represent  these  facts. 

(b)  Find  (1)  how  far  he  can  throw  when  the  angle  is  60°. 

(2)  what  angle  will  produce  a  throw  of  57  yds. 
•  (3)  what  is  the  greatest  distance  the  boy  can  throw, 
and  what  angle  will  produce  this. 

554.  The  vertices  of  a  pentagonal  field  are  located  by  the  fol- 
lowing points,  A  =  (-20,  15),  £=(10,  20),   C=(23,  -20),  D  = 
(_10,  -30),  #=(-30,  -10). 


THE  LOCUS  151 

(a)  Draw  the  outline  of  the  field. 

(b)  Give  new  values  to  A,  B,  C,  D,  E,  so  that  the  area  shall 
remain  the  same,  but  the  diagram  lie  wholly  in  the  first  quadrant, 
with  E  on  the  north-south  axis,  and  D  on  the  east-west  axis. 

(c)  Find  the  area  of  the  field. 

555.  The  boiling-point  of  water  on  a  Centigrade  thermometer 
is  marked  100°,  and  on  a  Fahrenheit  212°.    The  freezing-point  on 
the  Centigrade  is  zero,  and  on  the  Fahrenheit  is  32°.     Conse- 
quently a  degree  on  one  is  not  equal  to  a  degree  on  the  other. 

(a)  Show  that  the  correct  relation  is  expressed  by  the  equation 
C=l  (F~32),  where   C   represents  degrees   Centigrade,  and  F 
degrees  Fahrenheit. 

(b)  Construct  a  graph  of  this  equation.     Can  you,  by  means 
of  this  graph,  express  a  Centigrade  reading  in  degrees  Fahrenheit, 
and  vice  versa  f 

(c)  By  means  of  the  graph  express  the  following  Centigrade 
readings  in  Fahrenheit  readings,  and  vice  versa:     (1)   60°  C.; 
(2)  150°  F.;  (3)  -20°  C.;  (4)   -30°  F. 

(d)  What  reading  means  the  same  temperature  on  both  scales? 

c.  The  Graph  of  Equations. 

If  in  such  an  equation  as  x-\-y  =  10  various  values  of  x  are  taken 
as  abscissas  of  points  whose  ordinates  are  the  corresponding  values 
of  y,  and  the  points  are  joined,  we  have  what  is  known  as  the  graph 
of  the  equation.  It  is  a  fact  which  is  proved  in  more  advanced 
mathematics  that  the  graph  of  an  equation  of  the  first  degree  is 
always  a  straight  line. 

Thus,  if  we  represent  graphically  such  a  system  of  equations  as 
x+y  =  10,  and  x  -y  =  4,  we  have  two  straight  lines.  The  coordinates 
of  their  intersection  will  give  the  solution  of  the  equation.  Why? 

EXERCISES.     SET  LIV.     GRAPHIC  SOLUTION  OF  EQUATIONS 

Solve  the  following  systems  of  equations  graphically. 

556.  x+4y=ll  559.  2x-9y  =  2Z  562.  2x-3y  =  7 
2x-y  =  4                     5x+y=  -13  5x-7y=U 

557.  2z+3?/=19         560.  x+5y  =  Q  563.  6x-3y=15 
7x-2y  =  4:                   3x+9y=  -6 

558.  x+5y=  -3         561.  7x+2y=U 


152  PLANE  GEOMETRY 

II.  APPLICATION  OF  ELEMENTARY  GRAPHIC  MATHEMATICS  TO 

GEOMETRY 

Since  we  have  studied  the  graphic  solution  of  simultaneous 
equations,  the  idea  of  locus  (plural,  loci)  is  not  an  entirely  new  one. 
In  our  graphic  work  we  found  that  the  locus,  obeying  the  law 
expressed  by  a  linear  equation,  was  a  straight  line.* 

In  our  graphic  work  we  find  that  all  points  +2  units  from  the 
o>axis  are  to  be  found  on  a  line  parallel  to  the  z-axis  and  2  units 

above  it;  likewise  we  found  that 
all  points  in  this  line,  no  matter 
how  far  it  may  be  extended,  will 
be   +2   units   from   the  o>axis. 
Another  way  of  expressing  these 
V-'axis &    facts  is  to  say  that  the  path  of 
all  points  the  y-value  of  which  is 
2,  is  the  line  parallel  to  the  z-axis 
y  and  2  units  above  it,  and  next, 

that  the  7/-value  of  every  point  in 
the  line  parallel  to  the  x-axis  and  2  units  above  it  is  +2.  This  is 
stated  algebraically  by  means  of  the  equation  y  =  2. 

EXERCISES.     SET  LV.    THE  EQUATION  AS  THE  STATEMENT  OF 

A  LOCUS 

564.  Where  are  all  points  -2  units  from  the  z-axis  to  be  found? 
(Answer  in  a  complete  sentence.) 

565.  What  can  you  say  of  all  points  in  the  line  described  in 
your  answer  to  the  last  question?    (Answer  in  a  complete  sentence.) 

566.  State  the  law  which  is  obeyed  by  the  line  described  in 
exercise  564  by  means  of  an  algebraic  equation. 

567.  Where  are  all  points  + 10  units  from  the  ?/-axis  to  be  found? 

568.  What  can  you  say  of  all  points  in  the  line  described  in 
your  answer  to  the  last  question? 

569.  State  this  law  by  means  of  an  algebraic  equation. 

570.  Answer  the  last  three  questions,  inserting  the  following 
words  in  place  of  "+10  units  from  the  7/-axis": 

*  In  this  work,  those  for  whom  it  is  not  review  will  find  Auerbach,  An  Ele- 
mentary Course  in  Graphic  Mathematics,  Chapter  I  and  Chapter  III,  pp. 
22,  23,  and  28-31,  helpful. 


THE  LOCUS  153 

(a)    - 15  units  from  the  2/-axis.     (6)   +12  units  from  the  o>axis. 
(c)    —18  units  from  the  z-axis.     (d)    —7  units  from  the  y-axis. 
(e)    +13  units  from  the  2/-axis, 

571.  What  is  the  y-value  of  every  point  in  the  line  parallel  to 
the  z-axis  and  +6  units  from  it?     —6  units  from  it? 

572.  What  is  the  z-value  of  every  point  in  the  line  parallel  to 
the  7/-axis  and  +17  units  from  it?     -17  units  from  it? 

573.  What  is  the  path  of  every  point  whose  t/-value  is  +18? 
-18?    +20?     -3? 

574.  What  is  the  path  of  every  point  whose  x-value  is  30?     —27? 
+16?     -8? 

575.  Make  a  list  of  the  equations  expressing  the  facts  stated, 
in  order,  in  the  last  four  questions. 

576.  (a)  What  is  the  ?/-value  of  every  point  in  the  o>axis? 
(6)  What  is  the  path  of  every  point  whose  2/-value  is  this? 
(c)  What,  then,  is  the  equation  of  the  z-axis? 

577.  x  =  19  expresses  algebraically  what  two  facts? 

578.  What  is  the  equation  of  the  2/-axis?    Why? 

579.  What  is  the  equation  of  the  parallel  to  the  2/-axis  through 
the  point  (  —  5,  7)? 

The  arrangement  of  points  that  completely  fulfills  a  given  geometric 
condition  is  called  the  locus  of  that  condition.  This  arrangement 
usually  gives  rise  to  a  line  or  group  of  lines  either  straight  or  curved. 
For  instance,  the  locus  of  the  condition  expressed  by  the  equation 
x  =  7  is  the  line  drawn  parallel  to  the  2/-axis  at  a  distance  7  units 
to  the  right  of  it.  This  is  a  brief  way  of  saying  that  (1)  all  points 
in  this  line  are  7  units  to  the  right  of  the  ?/-axis  and  (2)  all  points 
7  units  to  the  right  of  the  2/-axis  lie  in  this  line. 

Because  of  the  idea  of  motion  involved,  another  acceptable 
definition  of  the  word  locus  would  be  :  The  complete  path  of  a 
point  that  moves  in  accordance  with  some  specified  geometric  condition. 
For  instance,  the  complete  path  of  a  point  that  moves  so  that  its 
distance  from  the  ?/-axis  is  -7  is  the  line  7  units  to  the  left  of  the 
2/-axis  and  parallel  to  it.  Hence  this  line  is  called  the  locus  of  the 
point  which  moves  so  as  to  remain  constantly  7  units  to  the  left 
of  the  i/-axis. 


154  PLANE  GEOMETRY 

EXERCISES.    SET  LV  (concluded) 

580.  What  is  the  locus  of  points: 

(a)  3  units  from  the  z-axis?          (6)    -5  units  from  the  2/-axis? 
(c)    —6  units  from  the  x-axis?      (d)  17  units  from  the  t/-axis? 

581.  What  two  facts  do  you  imply  in  the  answer  to  each  of  the 
parts  in  the  last  question? 

582.  Give  the  equation  expressing  the  condition  which  deter- 
mined each  of  the  loci  in  exercise  580. 

583.  What  is  the  locus  of  the  condition  expressed  in  each  of  the 
following  equations? 

(a)  z  =  15          (b)  y  =  -9          (c)  x=  -12          (d)  y  =  20 

584.  What  locus  is  represented  by  the  equation  x1  =  25? 

585.  What  locus  is  represented  by  the  equation 
(a)  x  =  yl  (d)  x-y=lW  (g) 
(6)  x=  -yt               (e)  x=  -3y?                (h) 
(c)  x+y  =  lW           (/)   x  =  3y7  (i) 


586.  Give  the  equation  of  the  locus  of  a  point: 

(a)  Just  as  far  from  the  £-axis  as  from  the  y-axis. 

(b)  Three  times  as  far  from  the  rr-axis  as  from  the  y-axis. 

(c)  Three  times  as  far  from  the  y-axis  as  from  the  x-axis. 

(d)  Minus  five  times  as  far  from  the  y-axis  as  from  the  z-axis. 

(e)  Minus  seven  times  as  far  from  the  x-axis  as  from  the  y-axis. 
(/)   Such  that  the  sum  of  its  distances  from  the  axes  is  —11. 
(g)  Such  that  three  times  its  distance  from  the  z-axis  increased 

by  9  times  its  distance  from  the  y-axis  is  26. 

(h)  Such  that  five  times  its  distance  from  the  z-axis  diminished 
by  twice  its  distance  from  the  y-axis  is  7. 

(i)  Such  that  the  sum  of  the  squares  of  its  distances  from  the 
axes  is  49. 

(j)  Such  that  four  times  the  square  of  its  distance  from  the 
z-axis  increased  by  the  square  of  its  distance  from  the  y-axis  is  144. 

Check  the  answer  to  each  part  of  the  last  two  questions  by 
plotting  the  equation. 

587.  The  theorem  of  Pythagoras  is  employed  to  find  the  "  equa- 
tion of  a  circle"  about  the  origin  as  a  center. 


THE  LOCUS  155 

Take  any  point  P  in  a  circle  about  the  origin  0.    Draw  the  ordi- 
nate  MP.     Let  OM=x,  and  MP=y.     Then 
If  the  radius  OP =r,  this  becomes  x2 -}-y2 
=r2.    This  equation  holds  for    the    co- 
ordinates of  any  point  on  the  circle,  and 
is  called  the  equation  of  the  circle,  r  being 
any  known  number. 

Form  the  equation  of  the  circle  with 
the  origin  as  center  and  (a)  7  as  radius, 
(b)  7\/2  as  radius. 

B.  THE  PECULIARITY  OF  THE   PROOF  OF  A  LOCUS 
PROPOSITION 

When  we  say  that  the  locus  of  points  on  this  page  just  one  inch 
from  its  right  edge  is  a  line  parallel  to  that  edge,  and  one  inch  in 
from  it,  we  really  imply  three  facts.  First,  that  any  point  on  that 
line  is  one  inch  from  that  edge;  second,  that  any  point  which  is 
one  inch  from  that  edge  and  on  the  page  is  on  that  line;  and  third, 
that  any  point  not  on  that  line  and  on  the  page  is  not  one  inch 
from  that  edge.  If  the  first  of  these  three  facts  be  called  the 
direct  statement,  we  already  know  that  the  second  is  its  converse. 
The  third  is  known  as  its  opposite. 

In  symbols,  three  theorems  so  related  might  be  stated  as  follows : 
Direct  theorem.  If  a  =  b,  then  c  =  d. 

Converse  theorem  If  c  =  d,  then  a  =  b. 

Opposite  theorem.  If  a^b,  then  c^d. 

Hence  we  see  that  while  the  converse  of  a  fact  simply  inter- 
changes its  data  or  what  is  given  with  its  conclusion,  the  opposite 
of  a  fact  negates  both  the  data  and  conclusion. 

Now  let  us  discover  what  we  can  as  to  the  truth  or  falsity  of 
converse  and  opposite  theorems  when  the  direct  theorem  is  true. 
EXERCISES.     SET  LVI.     DIRECT,  CONVERSE,  OPPOSITE 

The  following  exercises  will  help  us  in  this  task: 

588.  Form  (1)  the  converse,  and  (2)  the  opposite  of  each  of  the 
following  facts : 

(a)  If  a  man  lives  in  Boston,  he  lives  in  Massachusetts. 


156  PLANE  GEOMETRY 

(6)  If  it  rains,  the  ground  is  wet. 

(c)  If  two  lines  meet  at  right  angles  they  are  perpendicular  to 
each  other. 

(d)  All  vertical  angles  are  equal. 

(e)  The  supplements  of  equal  angles  are  equal,  or, 

If  two  angles  are  equal,  their  supplements  are  equal. 
(/)   All  men  are  bipeds. 

589.  (a)  Of  which  of  the  six  facts  mentioned  in  the  last  exercise 
are  the  converse  facts  true? 

(b)  Of  which  of  them  are  the  opposite  facts  true? 

(c)  Of  which  of  them  are  the  converse  facts  false — or  at  least 
not  necessarily  true? 

(d)  Of  which  of  them  are  the  opposite  facts  either  false  or  not 
necessarily  true? 

590.  (a)  Can  you  draw  any  conclusion  as  to  the  truth  or  falsity 
of  converse  and  opposite  theorems? 

(b)  Test  this  conclusion  with  several  more  instances. 
Though  a  statement  is  true: 

(1)  Its  converse  may  or  may  not  be  true. 

(2)  If  its  converse  is  true,  its  opposite  is  also  true. 

(3)  If  its  converse  is  false,  its  opposite  is  also  false. 
The  proof  of  this  fact  follows: 

Given:   That  when  a  =  b,  c=d,  and  when  c  =d,  a  =  b. 
Prove :  That  when  a  ^  b,  c  ^  d. 
Proof :  Suppose  c  =  d. 

Then  what  follows?    Why? 

What  conclusion  can  you  draw? 

Let  us  see  how  these  conclusions  help  us  to  decide  just  how  much  must  be 
proved  in  order  to  establish  the  truth  of  a  locus  proposition. 

Since  the  converse  of  a  fact  is  not  necessarily  true,  in  order  to  prove  a 
line  a  required  locus,  not  only  must  we  prove  (1)  that  any  point  in  it  fulfills 
the  required  conditions,  but  also  (2)  that  any  point  that  fulfills  the  required 
conditions  is  in  the  line.  It  is,  however,  unnecessary  to  prove  the  opposite, 
since  if  the  converse  has  been  proved  true,  we  know  the  opposite  is  also  true. 
In  short,  if  we  know  (1)  and  (2)  are  true,  we  know  without  further  proof  that 
any  point  not  in  the  line  does  not  fulfill  the  required  conditions. 
Suppose  we  wished  to  prove 

Theorem  40.    The  locus  of  points  equidistant  from  the  ends 
of  a  sect  isjhe  perpendicular  bisector  of  the  sect. 


P, 


THE  LOCUS  157 

Given:  XOY ±AOB  so  that  AO=OB. 

Prove:  XY  is  the  locus  of  points  equidistant  from  A  and  B. 

If  we  proved  first  that  any  point  P  in  -^ 

XY  is  equidistant  from  A    and  B,  and 
second  that  point  PI  which  is  equidistant  y 

from  A  and  B  lies  in  line  XY,  what  would  x/x 

we  know  about  any  point  not  in  X F?  XX 

.'.  (1)  Prove: PA=PB,  given:  AO^OB         /' 

and  P  in  XY _\_AB  at  0;  and  (2)_prove:   A^- 

OPlLAB,  given:  P±A  =PiB  and  AO=OB. 
To  prove  (1): 

What  parts  of  A  AOP  and  A  BOP  do 
you  know  are  equal? 

Are  the  A  congruent? 
To  prove  (2): 

What  parts  of  A  AOPi  and  A  BOPi  do  you  know  are  equal? 

Are  the  A  congruent? 

What  fact  must  you  prove  to  show  that  OP\±.ABl 

Is  there  any  other  converse  which  we  might  have  proved  in  place  of  the 
one  here  proved?    Why  are  there  two  converses  in  this  case? 

Cor.  1.    Two  points  each  equidistant  from  the  ends  of  a  sect 
fix  its  perpendicular  bisector. 

How  many  points  determine  a  straight  line? 

EXERCISES.     SET  LVII.     APPLICATIONS  OF  LOCUS 

591.  Show  why  a  circle  may  be  defined  as  the  locus  of  points 
at  a  fixed  distance  from  a  given  point. 

Describe  without  proof: 

592.  The  locus  of  the  tip  of  the  hand  of  a  watch. 

593.  The  locus  of  a  point  on  this  page  and  just  3"  from  the  upper 
right  corner. 

594.  The  locus  of  the  center  of  a  hoop  as  it  rolls  along  the  floor 
in  a  straight  line. 

595.  The  locus  of  the  edges  of  the  pages  of  a  book  as  it  is  opened. 

596.  The  locus  of  the  handle  of  a  door  as  it  is  opened. 

597.  The  locus  of  the  end  of  a  swinging  pendulum. 

598.  The  locus  of  places  described  as  1  mile  from  where  you  are 
standing. 

599*  The  locus  of  points  1'  above  a  given  shelf. 
d§00.  The  locus  of  points  1'  from  a  given  pbelf. 


158  PLANE  GEOMETRY 

601.  The  locus  of  the  center  of  a  circle  as  it  rolls  around  another 
circle,  its  circumference  just  touching  that  of  the  other  circle. 

602.  The  locus  of  the  center  of  a  ball  as  it  rolls  around  another 
ball,  its  surface  just  touching  that  of  the  other  ball. 

603.  The  locus  of  one  side  of  a  rectangle  as  it  revolves  about 
the  opposite  side  as  axis. 

604.  The  locus  of  the  entire  rectangle  in  the  last  exercise. 

605.  The  locus  of  a  point  at  3"  from  a  fixed  point  P. 

606.  The  locus  of  a  point  3"  from  a  given  line. 

607.  The  locus  of  a  point  equidistant  from  two  parallel  lines. 

608.  The  locus  of  a  point  equidistant  from  two  given  points. 

609.  The  locus  of  a  point  equidistant  from  two  intersecting  lines. 

610.  The  locus  of  a  point  the  distance  <-d— >  from  a  given  line  I. 

611.  The  locus  of  a  point  the  distance  <-d  ->  from  a  given  point  P. 

612.  The  locus  of  a  point  the  same  distance  from  the  center  and 
the  circumference  of  the  circle  c. 

We  shall  now  prove  one  more  veryimportant  fact  concerning  loci. 

Theorem  41.  The  locus  of  points  equi- 
distant from  the  sides  of  an  angle  is  the 
bisector  of  the  angle. 

I.  Given: %.CBA,  BXsp that  ^CBX  =  %.XBA. 
P  any  point  in  BX.    PR  _\_AB  cutting  AB  at  R. 
TSA.BC  cutting_BC  at  S. 

To  prove:  PR=PS. 

(Proof  left  to  the  student). 

II.  Given:   PI  a  point  within  the  %.CBA,  so 
that  P\R  (the  _|_  to  AB)  =/V?  (the  _L  to  BC). 

To  prove:  PiB  bisects  %.CBA. 

~S~    ~~G          (Proof  left  to  the  student.) 

Cor.  1.    The  locus  of  a  point  equidistant  from  two  intersecting 
lines  is  a  pair  of  lines  bisecting  the  angles. 

Hint:  At  what  angle  do  the  bisectors  of  any  two  adjacent  angles  formed 
by  the  pair  of  intersecting  lines  meet  each  other?  At  what  angle,  then,  do  the 
bisectors  of  the  vertical  angles  meet  each  other? 

EXERCISES.     SET  LVII  (concluded) 

613.  If  a  gardener  is  told  to  plant  a  bush  10'  from  the  north  fence 
does  he  know  exactly  where  to  plant  i t  ?  If  not,  state  another  direction 
which  might  be  given  him  that  he  may  know  just  where  to  plant  it. 


THE  LOCUS  159 

614.  Is  the  second  direction  given  the  gardener  the  only  one 
you  could  give  him  to  have  the  bush  definitely  located?    If  not, 
state  other  directions  which  might  have  answered. 

615.  How  many  loci  are  needed  to  locate  a  point  on  the  floor  of 
the  room?    On  any  one  of  its  walls?    On  the  ceiling? 

616.  (a)  Where  would  all  points  that  are  two  feet  from  the 
floor  of  a  room  lie? 

(6)  Where  would  all  points  3'  from  the  front  wall  lie? 

(c)  Where  would  all  points  that  are  both  2'  from  the  floor  and 
3'  from  the  front  wall  lie? 

(d)  Suppose  a  point  was  described  as  being  2'  from  the  floor, 
3'  from  the  front  wall,  and  4'  from  a  side  wall.    On  how  many  loci 
is  it?    Exactly  where  is  it? 

(e)  How  many  conditions  are  needed  to  fix  a  point  in  a  room? 

617.  A  man  wants  to  build  his  home  at  the  same  distance  from 
two  railroad  stations,     (a)  Is  the  location  of  his  home  fixed? 
(6)  If  at  the  same  time  he  wishes  to  build  a  half  mile  from  the 
bank  of  a  river  which  runs  parallel  to  and  four  miles  from  the 
road  connecting  the  stations,  is  the  location  of  his  home  fixed? 
Make   an  accurate   construction  showing  how   many  locations 
answer  the  description. 

618.  Prove  theorems  40  and  41  by  means  of  direct  and  opposite. 

619.  What  is  the  locus  of  the  vertices  of  triangles  which  have  a 
common  base  and  equal  areas? 

620.  What  is  the  locus  of  points  dividing  sects  which  connect 
a  given  point  and  a  given  line  in  the  ratio  of  5  to  7? 

621.  What  is  the  locus  of  the  vertices  of  triangles  resting  on  a 
common  base  and  having  fixed  areas  in  the  ratio  of  5  to  7? 

LIST  OF  WORDS  DEFINED  IN  CHAPTER  VII 

Locus,  opposite. 

SUMMARY  OF  THEOREMS  PROVED  IN  CHAPTER  VH 

40.  The  locus  of  points  equidistant  from  the  ends  of  a  sect  is  the  perpen- 
dicular bisector  of  the  sect. 

Cor.  1.    Two  points  each  equidistant  from  the  ends  of  a  sect  fix  its 
perpendicular  bisector. 

41.  The  locus  of  points  equidistant  from  the  sides  of  an  angle  is  the  bisector 
of  the  angle. 

Cor.  1.     The  locus  of  points  equidistant  from  two  intersecting  lines 
is  a  pair  of  lines  bisecting  the  angles. 


CHAPTER  VIII 

THE  CIRCLE 

We  have  not  only  used  the  word  "circle"  very  freely  throughout 
the  text,  but  have  also  used  our  compasses  for  the  construction  of 
the  circle,  or  any  part  of  it,  whenever  necessity  arose.  This  has 
been  due  to  the  fact  that  the  idea  of  a  circle  seems  to  be  one  with 
which  all  of  us  have  grown  up.  But  we  have  now  reached  the 
time  to  consider  the  idea  scientifically,  and  add  to  our  stock 
of  facts  concerning  it. 

Up  to  the  present  we  have  thought  of  the  circle  as  a  portion  of 
a  plane,  and  the  curve  bounding  it  as  its  circumference.  This  is 
not  the  sense  in  which  the  word  circle  is  used  as  we  advance  in 
mathematics,  so  we  shall  have  to  revise  our  notion  of  it.  The  word 
circle  is  used  to  refer  to  both  the  portion  of  a  plane  and  the  curve 
which  bounds  it  —  the  one  to  which  it  refers  being  determined  by 
the  context,  but  the  definition  covers  only  the  boundary.  When 
there  is  any  danger  of  ambiguity  the  word  circumference  will  be 
used  in  this  text. 

A  circle  is  a  plane  curve  which  contains  all  points  at  a  given 
distance  from  a  fixed  point  in  the  plane,  and  no  other  points*  Thus 
we  see  that  a  locus  definition  may  be  given  as  a  corollary  to  this 
one;  namely,  a  circle  is  the  locus  of  points  at  a  given  distance  from  a 
fixed  point*  The  fixed  point  is  called  the  center,  and  the  given  dis- 
tance (the  distance  from  the  center  to  any  point  on  the  circle) 
is  called  its  radius  (plural,  radii).  The  sect  through  the  center  and 
terminated  by  the  circle  is  called  its  diameter. 

At  this  point  we  may  state  several  corollaries  to  these  definitions. 
It  will  be  left  to  the  student  to  verify  them. 

^  Cor.  1.    All  radii  of  equal  circles  are  equal. 
-  Cor.  2.    Circles  of  equal  radii  are  equal. 

.  3.    All  diameters  of  equal  circles  are  equal. 


*  Refer  to  Exs.  585  (i),  586  (i),  588,  591,  and  611  for  previous  illustrations 
of  this  definition. 

160 


THE  CIRCLE  161 

Cor.  4.    A  point  is  inside,  on,  or  outside  a  circle,  according  as 

its  distance  from  the  center  is  less  than,  equal  to,  or 

greater  than  the  radius. 
Cor.  5.    A  point  is  at  a  distance  less  than,  equal  to,  or  greater 

than  the  radius  from  the  center  according  as  it  is  inside, 

on,  or  outside  the  circle. 

A.  PRELIMINARY  THEOREMS 
^   Theorem  42.     Three  points  not  in  a 
straight  line  fix  a  circle. 
Given:   Points  A,  B,  D,  not  in  a  straight  line. 
To  prove:    (1)  A  circle  can  be  passed  through  A, 
B,D. 

(2)  Only  one  circle  can  be  passed  through  A, 
B,D. 

Suggestions  for  proof:  What  is  the  locus  of  the  -D 

centers  of  all  circles  passing  through  A  and  D?    Why?    Through  B  and  D? 
By  means  of  the  transversal  XY  in  the  diagram,  show  that  PY  and  QX 

are  not  parallel,  and  that  hence  point  C  exists 
Why  can  no  second  point  such  as  C\  exist? 

EXERCISE.  SET  LVIII.  THE  CIRCLE  AS  A  LOCUS 
622.  An  amusement  park  is  to  be  located  at  the  same  distance 
from  each  of  three  villages  V\,  V2,  and  V3.  V2  is  5  miles  from  Vi, 
Vs  is  6  miles  from  Vi,  and  V3  is  8  miles  from  V2.  Show  by  an  accur- 
ate construction  the  actual  location  of  the  park.  Can  you  think 
of  any  locations  of  V\,  V2,  and  at  the  same  time  F3  which  would 
make  it  impossible  to  have  a  park  so  located?  (Use  sect  V2  V3io 
represent  8  mi.)  V2 — —  -F3 

In  the  accompanying  diagram,   ^BOA  is  known  as  a  central 
angle.    Define  such  an  angle. 

Any  portion  of  a  circle,  such  as  BA  is  known 
as  an  arc.  When  referring  to  any  definite  arc, 
such  as  BA  or  CD,  we  write  it  thus:  BA,  CD. 
|^  Any  two  points  on  a  circle   (unless  they  are 
the  ends  of  a  diameter)  are  the  ends  of  two 
arcs  knownjis  minor  and  major  arcs.^For 
instance,  CXD  is  the  minor  CD,  and  DYC  is 
the  major  DC.     The  shorter  of  two  arcs  cut  off  by  any  two  points  on 
11 


162 


PLANE  GEOMETRY 


a  circle  is  known  as  the  minor  arc,  and  the  longer  is  known  as  the 
major.  When  not  otherwise  stated ,  the  minor  arc  is  referred  to. 

The  sect  CD  is  known  as  a  chord .  Define  the  word  chord.  The 
diameters  of  a  circle  are  simply  chords.  Why? 

Angles  in  a  circle  are  said  to  intercept  arcs,  and  chords  are  said 
to  subtend  arcs.  Intercept  comes  from  the  Latin  "inter,"  meaning 
"between,"  and  "capio,"  meaning  "to  take,"  hence  the  angle 
intercepts  or  "takes  the  arc  between  its  sides."  Subtend  comes 
from  the  Latin  "sub,"  meaning  "under,"  and  "tendere,"  meaning 
"to  stretch."  Thus,  as  we  see,  the  chord  is  the  straight  line  which 
"stretches  under  the  arc."  Arcs  are  considered  passive,  and  are 
referred  to  as  being  "intercepted  by"  angles,  and  "subtended  by" 
chords,  although  in  engineering  the  expression  "the  central  angle 
subtended  by  an  arc  of  n°  "  is  very  common.  If  the  expression  is* 
used  in  this  text  it  will  only  be  when  referring  to  engineering 
problems. 

Theorem  43.  In  equal  circles  equal  central  angles  intercept  equal 

arcs,  and  conversely. 

I.  Direct. 
Given:    OC  =  Od 


Method  of  proof,  superposition, 
make  coincide  in  proving  the  direct? 


Prove:  AB^XY. 
II.  Converse. 

Given:    0C=  0d,  AB=XY. 

Prove:    X,.ACB  =  %.XCiY. 
In  superposing,  which  parts  will  you 
Which  in  proving  the  converse? 


B.  THE  STRAIGHT  LINE  AND  THE  CIRCLE 

Theorem  44.  In  equal  circles,  equal  arcs  are  subtended  by  equal 
chords,  and  conversely. 

Suggestion:    Prove  A  ACB^ 


What  means  have  you  of 
proving  the  triangles  con- 
gruent in  the  direct? 
What  in  the  converse? 


THE  CIRCLE 


163 


EXERCISES.    SET  LIX.    CONGRUENCE  OF  CURVILINEAR 

FIGURES 
623.  Prove  that  the  curved  figures 


* 


G 


BDEF  and  CFGH  are  congruent.  The 
figure  is  based  on  a  network  of  equi- 
lateral triangles.  The  vertices  are  the 
centers,  and  the  sides  the  radii  for  the 
arcs. 

624.  In  the  accompanying  figure 
prove  that  the  curved  triangles  ABC, 
CUD,  etc.,  are  congruent.  Also  BCDEFG  and  DHML. 


H 


Y  Theorem  45.    A  diameter  perpendicular  to  a  chord  bisects 
it  and  its  subtended  arcs. 

Given:    QO,  diameter^ EF^L  chord  AB  at  D. 
Prove:  AD=DB.    AE=EB.    BF=fA. 
Proof:  AE=EB  if  what  angles  are  equal? 

These  angles  are  equal  if  what  triangles  are  con- 
gruent?   Write  a  complete  proof.         ^ 

By  means  of  what  angles  can  you  prove  BF 

Cor.  1.    A  radius  which  bisects  a  chord  is 
perpendicular  to  it. 

Which  of  the  methods  of  proving  lines  perpendicular  can  be  applied  here? 

Cor.  2.    The  perpendicular  bisector  of  a  chord  passes  through 
the  center  of  the  circle. 

Hints:   Draw  the  radius  which  bisects  the  chord  and  prove  the  given 
bisector  coincident  with  it,  or  treat  as  a  locus. 


164 


PLANE  GEOMETRY 


EXERCISES.    SET  LX.    CONSTRUCTIONS  BASED  UPON  CIRCLES 

625.  Give  the  con- 
struction for  Fig.  1.  ° 


From  Carlisle  Cathedral. 
(Figs.  1  and  2  applied.) 

626.  Inscribe  Fig.  1  in  a  given  circle.    (See  Fig.  2.) 

627.  Give  the  construction  for  the  design  shown  in  Fig.  3. 

Suggestion :  Construct  the  equilateral  AABC.  Divide  each  side  into  three 
equal  parts  and  join  the  points  as  indicated.  The  intersections  are  the  centers, 
and  AM  is  the  radius  for  the  arcs  drawn  as  indicated. 


From  Exeter  Cathedral. 
(Fig.  3  applied.) 


628.  Construct  Fig.  IV. 

Suggestion :  Construct  the 
equilateral  AABC.  A,J1, 
and  C  are  the  centers  for  CB, 
CA,  and  AH.  The  semi- 
circles are  constructed  on  the 
sides  of  the  linear  triangle 
as  diameters. 


THE  CIRCLE 


165 


629.  A  civil  engineer  wishes  to  continue  the  circular  track  AB 
for  some  distance.     Suggest  how  he 

can  do  it. 

630.  From  the  measurements  of  a  A 
piece  of  broken  wheel  a  new  wheel  is  to 
be  cast  of  the  same  size.    Show  how 
to  find  the  radius  of  the  new  wheel. 

V  Theorem  46.    In  equal  circles,  equal  chords  are  equidistant 

from    the    center,    and 
conversely. 

I.  In  the  direct  what 
parts  are  known  to  be 
equal  in  AOBX  and 


prove 


.     II.  State    and 
the  converse. 

EXERCISE.    SET  LXI.    EQUAL  CHORDS 


631.  The  following  method  of  locating  points  on  an  arc  of  a 

circle  that  is  too  large  to  be  described  by  a  tape  is  used  by  engineers. 

If  part  of  the  curve  APB  is  known, 

take  P  as  the  mid-point.     Then 

stretch  the  tape  from  A  to  B  and 

draw  PM  perpendicular  to  it.  Then 

swing  the  length  AM  about  P,  and 

PM  about  B,  until  they  meet  at  L, 

and  stretch  the  length  AB  along  PL  to  Q.  This  fixes  the  point  Q. 
In  the  same  way  fix  the  point  C.  Points 
on  a  curve  can  thus  be  fixed  as  near  to- 
gether as  we  wish.  Why  is  this  method 
correct? 

A  straight  line  is  said  to  be  tangent  to  a 
circle  when  it  touches  it  once,  and  only  once. 
Thus  XY  is  tangent  to  QC  if  it  touches 
at  pt.  P,  and  nowhere  else. 


166 


PLANE  GEOMETRY 

V 

^Theorem  47.    A  line  perpendicular 

to  a  radius  at  its  outer  extremity  is 
tangent  to  the  circle. 

Given:    OO,  AB  J_  radius  OC  at  C. 
Prove:  AB  tangent  to  00. 


PROOF 


(1)  Given. 

(2)  Authorities  left  to  the  student  to 

insert. 


(1)  AB  J.  radius  at  C. 

(2)  .-.  AB  touches  OO  at  C. 

(3)  It  remains  to  prove  that  no  other 

point  such  as  P  in  AB  touches 
_OO._/.  Draw  sect  OP. 

(4)  OC^L  AB^_  (4) 

(5)  .'.OPjLAB.  (5) 

(6)  .*.  OP  >  OC.  (6) 

(7)  /.  P  lies  outside  of  00.  (7) 

(8)  /.  CVand  only  C  in  AB  is  in  OO.     (8) 

(9)  .'.  AB  is  tangent  to  OO.  (9) 

The  first  three  corollaries  following  are  partial  converses  of  this 
theorem. 

We  are  not  yet  ready  to  prove  the  converse  of  Theorem  6. 
It  will  be  proved  later  under  the  topic  of  inequalities.  For  the 
present,  then,  we  shall  add  it  as  a  postulate. 

Postulate,  the  third  postulate  of  perpendiculars: 

The  shortest  distance  from  a  point  to  a  line  is  the  perpendicular 
to  that  line. 

COT.  1.    A  tangent  to  a  circle  is  perpendicular  to  the  radius 

drawn  to  the  point  of  contact. 
Given:  Xf  tangent  to  OO  at  P,  radius  UP. 
Prove:  XY  ±OP. 
Suggestions  f  or  proof  :  Draw  OQ  to  any  point  QmXY. 

Where  doesQ  lie? 

How  does  OQ  compare  with  OP? 

What  conclusion  can  be 
drawn  with  respect  to  OP? 

Cor.  2.  The  perpendicular  to  a  tangent  at 
the  point  of  contact  passes  through 
the  center  of  the  circle. 

Hint:  Show   that  KP  coincides  with  the 
radius  CP,  and  .'.  C  lies  on  KP. 


- 


- 


THE  CIRCLE 


167 


*•* 


Cor.  3.    A  radius  perpendicular  to  a  tangent  passes  through  the 
point  of  contact. 

Hint:   Draw  the  radius  to  the  point  of  con- 
tact P,  and  show  that  CR  coincides  with  CP. 

Cor.  4.  Only  one  tangent  can  be 
drawn  to  a  circle  at  a  given 
point  on  it. 

Hint:  How  many  ±s  can  be  erected  to  CR 
atjR? 

EXERCISES.     SET  LXII.     THE  TANGENT  AND  THE  CIRCLE 
632.  When  a  ray  of  light  strikes  a  spherical  mirror  (represented 
in  cross  section  by  the  arc  of  a  circle),  the  angle  of  incidence  is 
found  by  drawing  a  tangent  to  the  circle  at  the 
point  of  incidence,  and  erecting  a  perpendicular  to 
the  tangent  at  that  point.    In  this  case  the  perpen- 
dicular (called  the  normal)  is  a  diameter.    Why? 
d633.  The  line  of  propagation  of  a  sound  wave 
also  follows  the  law  of  reflection  of  a  ray  of  light, 
namely,  that  the  angle  of  incidence  is  equal  to  the  angle  of  reflection. 
The  circular  gallery  in  the  dome  of  St.  Paul's  in  London  is  known 
as  a  whispering  gallery,  for  the  reason  that  a  faint  sound  produced 
at  a  point  near  the  wall  can  be  heard  around  the  gallery  near  the 
wall,  but  not  elsewhere.    The  sound  is  reflected  along  the  circular 
wall  in  a  series  of  equal  chords.    Explain  why  these  chords  are  equal. 

634.  What  is  the  locus  of  the  centers  of  a  number  of  hoops  of 
different  sizes  (one  inside  the  other)  tied  together  at  one  point? 

635.  What  is  the  locus  of  the  centers  of  all  circles  tangent  to  a 
given  line  at  a  given  point? 

636.  What  is  the  locus  of  the  center  of  a  wheel  as  it  rolls  straight 
ahead  along  level  ground?    Prove  this  fact. 

637.  What  is  the  locus  of  the  centers  of  all  circles  tangent  to 
both  sides  of  an  angle? 

638.  Two  straight  roads  of  different 
width  meet  at  right  angles.  A  is  the  nar- 
rower, B  the  wider.    It  is  desired  to  join 
them  by  a  road  the  sides  of  which  are 
arcs  of  circles  tangent  to  the  sides  of  the 
straight  roads.    What  construction  lines 
are  necessary?    Draw  a  figure. 


168 


PLANE  GEOMETRY 


639.  How  would  you  construct  a  tangent  to  a  given  circle  at  a 

given  point  P  on  the  circle? 

Theorem  48.  Sects  of  tan- 
gents from  the  same  point 
to  a  circle  are  equal. 

Why  are  A  OPX  \nd  OPY 
congruent? 

Circles  are  said  to  be  tangent  to  each  other  when  they  are  tangent 
to  the  same  line  at  the  same  point.    Why  is  OCi  tangent  to 
Why  to  QC3?    Why  is  QC2  tangent  to  OC3? 

Circles  are  externally 
tangent  when  their  centers 
lie  on  opposite  sides  of  the 
common  tangent.  Name 
two  pairs  of  circles  that 
are  externally  tangent 
in  the  diagram. 

Circles  are  internally 
tangent  when  their  centers  lie  on  the  same  side  of  the  common  tangent. 
Which  circles  in  the  diagram  are  internally  tangent? 

EXERCISES.    SET  LXIIL     TANGENT  CIRCLES 

640.  (a)  How  are  the  hoops  mentioned  in  Ex.  634  related  to  one 
another? 

(6)  How  is  the  line  in  which  their  centers  lie  related  to  their 
common  tangent?    WTiy? 

(c)   Is  this  fact  true  of  the  centers  of  two  cog-wheels  when  they 
A  mesh?    Why? 

The  line  of  centers  of 
two  circles  is  the  sect  con- 
necting their  centers. 

Theorem  49.  The  line 
of  centers  of  two  tangent 
circles  passes  through 
their  point  of  contact. 

(For  suggestions  see  Ex.  634,  p.  167,  and  Ex.  640,  p.  168.) 


THE  CIRCLE 


169 


The  common  chord  of  two  intersecting  circles  is  the  sect  connecting 
their  points  of  intersection. 

EXERCISES.     SET  LXIII  (concluded) 

641.  How  is  the  line  of  centers  of  two  intersecting  circles  related 
to  their  common  chord  ?    Prove  your  answer. 

642.  If  two  cog-wheels  mesh,  show  that  the  point  where  they 
mesh  is  in  a  straight  line  with  the  centers  of  the  wheels. 

643.  (a)  Show  how  to  construct  an 
equilateral  Gothic  arch.  (See  the  accom- 
panying diagram.) 

Suggestion  :  Construct  the  equilateral  tri- 
angle ABC.  With  A  and  B  as  centers  and  AB 
as  radius,  construct  the  arcs  BC  and  AC. 

(6)  If  E,  F,  and  D  are  the  mid-points 
of  the  lines  AC,  CB,  and  AB,  respec- 
tively, prove  that  equal  equilateral  tri- 
angles  are  formed. 

(c)  Construct  the  equilateral  arches  ADE  and  DBF  and  the 
curved  triangle  EFC. 

Suggestion:   Points  D,  E,  C,  F,  B,  and  A  are  the  centers  and  AD  is  the 
radius  for  the  arcs  drawn  as  indicated. 

d644.  An  application  of  geometry  to  engineering  is  seen  in  cases 
where  two  parallel  streets  or  lines  of  track  are  to  be  connected  by  a 

"reversed  curve."  If  the  lines 
are  AB  and  CD,  and  the  con- 
nection is  to  be  mad^  from 
B  to  C,  as  shown,  we  may  pro- 
ceed as  follows:  Draw  BC 
and  bisect  it  at  M .  Erect 
PO  the  perpendicular  bisector  of  BM,  and  BO  perpendicular  to 
AB.  Then  0  is  one  center  of  curvature.  In  the  same  way  fix  0\. 
The  curves  may  now  be  drawn,  and  they  will  be  tangent  to  AB,  to 
CD,  and  to  each  other.  Prove  that  the  curve  BMC  is  a  reversed 
curve  tangent  to  AB  and  CD;  i.e.,  prove  (a)  BM  tangent  to  AB 
at  B^CM  tangent  to  CD  at  C;  (b)  BM  tangent  to  CM  at  M] 
(c)  BM  =  CM. 


170 


PLANE  GEOMETRY 


645.  The  figure  represents  a  Persian  arch.  Triangles  ABC  and 
DEF  are  congruent  and  equilateral.  The  centers  of  the  upper 
arcs  MC  and  NC  are  respectively  D  and  F\ 
while  the  lower  arcs  are  drawn  with  the  cen- 
ter E.  Prove  that  the  area  of  the  arch  equals 
the  area  of  triangle  ABC. 

646.  The  trefoil  ADBF,  etc.,  is  constructed 
from  circles  described  on  the  semisides  of 
A  ABC.  The  points  D,  E,  and  F  are  the 
centers  for  the  arcs  which  are  tangent  to  the  sides  of  AABC,  and 
which  form  the  trefoil  HYKZGX.  If  PD  and  RF  in  Fig.  1  are 
radii  for  the  arcs  ZK  and  YK,  prove  that  PD  =  FR. 


FIG.  1. 


FIG.  1.  applied. 


d647.  The  semicircle  AGHB  in  Fig.  2  is  constructed  on  AB  as 
diameter,  and  CD  is  perpendicular  to  AB  at  its  mid-point. 

(a)  Construct  arcs  CH  and 
CG  tangent  to  the  line  CD  at 
point  C,  and  to  the  semicircle. 

Suggestion  :  Make  KC  =  AD. 
Draw  NE,  the  perpendicular  bisector 
of  KD,  meeting  CK  extended  at  E. 
E  is  the  center  for  the  arc  CH.  What 
general  problem  in  construction  of 
circles  is  involved  here? 

(6)  If  the  arcs  CH,  drawn 
with  E  as  a  center,  and  HB  drawn  with  D  as  a  center,  are  tangent 
at  H,  prove  that  the  points  D,  H,  and  E  are  collinear. 


A  D  B 

FIG.  2.    Three-centered  ogee  archea. 


THE  CIRCLE 


171 


(c)  Prove  that  C,  H,  and  B  are  collinear. 

Suggestion:    Join  C  and  H,  and  B  and  C,  and  prove  that  each  is  parallel 
to  DK. 

(d)  If  AB  =  8,  and  CD  =  8,  find  the  length  of  CE. 

Suggestion  :  KD  =  4  A/5.    Compare  the  sides  of  the  similar  triangles 
KCD  and  KNE.    Find  KE  and  hence  C#. 


(e)  IfAB  =  s  and  CZ)  =  /i,  find  the  length  of  CK. 

648.  What  must  be  the  relation  between 
the  length  and  width  of  the  rectangle 
A  BCD  in  order  that  the  tangent  circles 
may  be  inscribed  as  shown. 


Stair  Railing. 


649.  ABCD  is  a  parallelogram  with  four  tangent  circles  inscribed. 

(a)  If  the  lines  AC  and  BD 
are  supposed  to  be  indefinite  in 
extent,    show  how  to  construct 
circle  0  tangent  to  the  lines  A  C, 
AB,  and  BD,  and  circle  X  tan- 
gent to  lines  AC  and  BD  and  to 
circle  0. 

(b)  If  E  is  the  mid-point  of  AB, 
and  0  and  X  are  the  centers  of 
the  circles,  prove  that  the  points 
E,  0,  and  X  are  collinear. 

650.  Fig.  1  shows  a  trefoil 
formed  of  the  three  circles  X,  Y,  and  Z  tangent  to  each  other  at 
the  points  T,  S,  and  R.  It  is  E 

inscribed  in  the  circle  as  shown.     ^~~ /^T^\ 13t 

(d)  Show  how  to  construct  the 
figure. 

Solution  :  Circumscribe  an  equilateral 
triangle  about  the  circle.  Connect  each 
vertex  with  the  center.  Inscribe  a  circle 
in  each  of  the  triangles  FOG,  GOE,  and 
EFO. 

(b)  Prove  that  the  small  circles 
are  tangent  to  the  large  circle  and 

tO  each  Other.  FIG.  I.      Trefoil  fcrmed  cf  tangent  circlea- 


172 


PLANE  GEOMETRY 


651.  (a)  Construct  the  quadrifoil  of  tangent  circles  inscribed 
in  a  square  so  that  each  circle  is  tangent  to  two  sides  of  the  square 
and  to  two  other  circles. 


E  -°  Inlaid  tile  design. 

FIG.  2.  (Fie.  2  applied.) 

(6)  Prove  that  the  lines  joining  the  centers  of  the  circles  form 
a  square. 

C.  THE  ANGLE  AND  ITS  MEASUREMENT 
v  Theorem  50.    In  equal  circles  central  angles  have  the  same 
ratio  as  their  intercepted  arcs. 

Given:  OC=JDCi,  %.ACB&nd 
4  XCiY,  AB  and  XY  com- 
mensurable. 


Prove: 

Suggestions  for  proof:  Select  a 
unit  of  measure  for  AB  and 
XY 
Divide  these  arcs  into  such  units  and  connect  the  points  of  division  with 

the  centers  C  and  C\. 

What  can  you  say  of  all  the  central  angles  thus  formed? 
How  do  %.s_BCA  and  YCiX  compare? 
How  does  BA  compare  with  YX* 

Cor.  1.    A  central  angle  is  measured  by  its  intercepted  arc. 
Suppose  4.  a  were  the  angular  unit  of  measure,  and  a  the  circular  unit  of 
measure  in  a  circle  of  radius  r. 

What  would  be  the  numerical  measure  o 
What  would  be  the  numerical  measure  o 
How  do  these  numerical  measures  compare? 


THE  CIRCLE 


173 


Up  to  the  present  time  we  have  emphasized  the  fact  that  a 
magnitude  can  be  measured  by  a  unit  of  the  same  kind  only.  We 
must  then  justify  a  statement  such  as  that  given  in  Cor.  1,  Theorem 
50.  This  corollary  should  be  stated  as  follows:  The  ratio  of  a 
central  angle  of  a  circle  to  the  angular  unit  is  equal  to  the  ratio  of 
its  intercepted  arc  to  the  circular  unit  ;  or  :  The  numerical  measure 
of  a  central  angle  of  a  circle  is  equal  to  the  numerical  measure  of 
its  intercepted  arc.  But  since  this  fact  is  one  frequently  referred 
to,  and  the  correct  statement  of  it  is  so  lengthy,  mathematicians 
have  agreed  to  the  abbreviated  statement  given  in  Cor.  1.  The 
symbol  "  2£  "  is  used  for  "is  measured  by."  It  suggests  the  ideas 
of  both  equality  and  variation. 

A  secant  is  a  straight  line  which  intersects  the  circle. 


EXERCISES.    SET  LXIV.    SECANT  AND  CIRCLE 


=  49, 


§652.  (a)  Show  by  a  graphic  solution  of  the  equations  x2 
and  x  =  3,  that  a  secant  cuts  a  circle  in  two  points. 

(b)  What  numbers  would  have  to  replace  3  in  the  second  equa- 
tion to  change  the  equation  to  one  of  a  tangent? 


)f  Theorem  51.    Parallels  intercept  equal  arcs  on  a  circle. 

p — ^T^ — Q  x  „_       T 


Q 


Case  I.    When  the  parallels  are  a  secant  and  a  tangent.    (Fig.  1.) 
Given:  PQ  tangent  to  circle  C  at  T,  secant  RS  \\  PQ,  cutting  C  at  R  and  S. 
Prove:  ST  =  TR. 
Suggestions  for  proof  :  Draw  diameter  through  T,  cutting  QC  in  M. 

What  relation  exists  between  TCM  and  PQ? 

Then  what  relation^exists  between  TCM  and  ~RSt 

What  follows  as  to  ST  and  TRt 


174  PLANE  GEOMETRY 

Case  II.    When  both  parallels  are  secants.    (Fig.  2.) 
(Proof  left  to  the  student.)    Suggestion :  Draw  the  diameter  perpendicular 
to  one  secant. 

Case  III.    When  both  parallels  are  tangents.    (Fig.  3.) 
Draw  the  diameter  through  T,  and  give  the  proof  in  full. 
EXERCISES.    SET  LXV.     CIRCLES 

653.  Does  it  make  any  difference  in  what  order  the  cases  under 
Theorem  51  are  proved,  if  the  proofs  are  given  as  suggested? 

654.  Can  you  suggest  methods  of  proof  for  Cases  II  and  III 
which  depend  upon  Case  I?  Which  methods  do  you  prefer?  Why? 

655.  Construct  a  diagram  consisting  of: 

(a)  Two  concentric  circles  whose  radii  are  in  the  ratio  of  1  to  3. 
(6)  Six  circles  lying  between  them,  tangent  to  them,  and  each 
tangent  to  two  others. 

656.  Make  a  diagram  01  the  mariner  s  compass,  putting  in  six- 
teen points  of  the  compass. 

Angles  such  as  ABC  in  Figs.  1,  2,  3,  Theorem  52,  are  called 
inscribed  angles.  Their  vertices  are  not  only  in  the  circle,  but  their 
sides  are  chords.  An  angle  is  said  to  be  inscribed  in  a  semicircle 
when  its  sides  intercept  a  semicircle,  in  less  than  a  semicircle  when 
its  sides  intercept  a  major  arc,  and  in  more  than  a  semicircle  when 
its  sides  intercept  a  minor  arc. 

Since  measurement  is  but  a  numeric  relation,  axioms  may  be 
converted  into  authorities  for  statements  concerning  measure- 
ment by  simple  changes  such  as  those  illustrated  by  the  following: 

The  measure  of  the  sum  of  two  magnitudes  of  the  same  kind  is 

If  X  OC  ft  ) 

equal  to  the  sum  of  their  measures.      E.g. :       ,    ~     >  x  -\-y^a-\-b. 

The  measure  of  any  multiple  of  a  magnitude  is  equal  to  that 
multiple  of  its  measure.  E.g.:  If  x<£a,  then 
mx^ma. 

Theorem  52.    An  inscribed  angle  or  one 
formed  by  a  tangent  and  a  chord  is  measured 
by  one-half  its  intercepted  arc. 
Given:    QO,  4. ABC  so  that  B  is  on  the  circle. 
To  prove:    ^.ABC^.^  its  intercepted  arc. 

FIQ  1  Suggestions:   Fig.  1.  Compare  4 ABC  with  %.AOC. 

What  is  the  measure  of  2^. ABC? 


CIRCLE 


175 


Figs.  2  and  3.    By  drawing  the  diameter  through  B,  reduce  these  cases 

to  that  of  Fig.  I. 
Fig.   4.    DrawZAF||DC. 

What  is  the  measure  of  2(.YABt 

What  arc  may  be  substituted  for  YBl    Why? 

What,  then,  is  the  measure  of  ^.DBAl    Of  its  supplement  2$. ABC1 

B 

D B 


FIG.  2. 


FIG.  4. 


EXERCISES.    SET  LXVI.    INSCRIBED  ANGLES 

657.  What  kind  of  angle  is  inscribed  in  a  semicircle?    In  less 
than  a  semicircle?     In  more  than  a  semicircle? 

658.  In  carpentry,  circular  pieces  of  molding  for  door  panels, 
etc.,  are  sometimes  turned  out  in  the  form  of  rings  on  a  lathe; 
then  these  are  cut  into  pieces  according 

to  the  places  in  which  they  are  to  be  used. 

To  cut  such  a  ring  into  two  equal  parts, 

place  a  carpenter's  square  upon  it  with 

the  heel  at  the  edge  of  the  ring,  and 

mark  the  points  A   and  C  where  the 

arms  of  the  square  cross  the  edge  of  the  ring. 

half  of  the  ring. 

659.  Pattern-makers  and  others  use 
the  carpenter's  square  as  follows  to 
determine  if  the  " half-round  hole"  is 
a  true  semicircle :  The  square  is  placed 
as  in  the  figure .  If  the  heel  of  the  square 
touches  the  bottom  of  the  hole  in  all 

positions  of  the  square,  while  the  sides  rest  against  the  edges  of 

the  hole,  the  hole  is  a  true  semicircle.    Justify  this  test. 

660.  Prove  that  the  semicircles  intersect  in  pairs  on  the  sides 

of  the  triangle  in  Ex.  628. 


Show  that  ABC  is 


176 


PLANE  GEOMETRY 


661.  In  practical  work  a  line  EF  is  sometimes  drawn  parallel 

,__j  ^-4--^  to  a  given  line^4.£,  as  shown  in 

the  figure.  Explain  the  con- 
struction, and  prove  that 
EF\\AB. 

^  \[  XY  \{''  jf  p  662.  A  ship  is  steered  past 
XN-— -''  s^»_-x'  a  known  region  of  danger  as 

follows :  A  chart  is  made  in  which  a  circle  is  drawn  through  two 

points  A  and  B,  which  can  be  seen  from  the 

ship,  and  with  sufficient  radius  to  inclose  the 

danger  region.    The  inscribed  angle  ACB  is 

measured.    Observations  of  A  and  B  are  made 

from  the  ship  from  time  to  time,  and  the  course 

of  the  ship  directed  so  that  the  angle  between 

the  directions   to  A   and   B  never  becomes 

*  T 

greater  than    -£ACB.    Justify  this  method. 

663.  Show  by  a  diagram  how  you  can  use  a  protractor  with  a 

plumb-line  attached  to  determine  the  horizontal  line  AC  while 
sighting  the  top  B  of  a  building. 

664.  The  diagram  shows  how  the  latitude 
of  a  place  may  be  determined  by  observa- 
tion of  the  pole  star.  Let  EEi  represent 
the  equator,  AAi  the  axis  of  the  earth,  P  the 
place  whose  latitude  is  to  be  found,  PF  a 
plane  (the  horizon)  tangent  to  the  earth  at 
P,  and  PS  the  line  of  observation  of  the 
pole  star.  Then  <£  a  represents  the  latitude, 
is  called  the  altitude  of  the  pole  star.  Ap 

For  practical  purposes  we  may  assume  that 

AAi  ||  PS.    Prove  that  <£«i  =  ^a. 

665.  Suppose  XY  to  be  the  edge  of  a 

sidewalk,  and  P  a  point  in  the  street  from 

which  we  wish  to  lay  a  gas  pipe 

perpendicular  to  the  walk.  From   -2T — 

P  swing  a  cord  or  tape,  say  50  feet  long,  until  it  meets  XY  at  A. 

Then  take  M,  the  mid-point  ©f  PA,  and  swing  MP  about  M,  to 

meet  XY  at  B.   Then  B  is  the  foot  of  the  perpendicular.  Verify  this. 


B 


THE  CIRCLE 


177 


666.  By  driving  two  nails  into  a  board  at  A  and  B,  and  taking 
an  angle  P  made  of  rigid  material,  a  pencil 
placed  at  P  will  generate  an  arc  of  a  circle 
if  the  arms  slide  along  A  and  B.    Why? 
Try  the  experiment. 

d667.  Circle  O  is  tangent  to  the  sides  of 
square  ABCD  at  points  E,  F,  G,  and  H, 
and  cuts  the  diagonals  at  points  K,  L,  M,  and  N.  Point  X  on  OE 

D O  _c         is  so  chosen  that  KX=KE.  OX 

=OY=OZ=OW,tm&  the  points 
are  joined  as  indicated. 


H 


A-  E  B  Parquet  flooring.  Arabic  and  Roman. 

(a)  Prove  that  KELFM,  etc.,  is  a  regular  octagon,  and  that 
KELX,  LYMF,  and  MGNZ  are  congruent  rhombuses. 
(6)  Prove  that  XO=AK. 
Suggestion:  Compare  &KXO  and  AEK.    ^  =  ^.5  =  1  rt.2$.. 


EXERCISES. 


Theorem   53.    An  angle  whose  vertex  is 
B   inside  the  circle  is  measured  by  half  the  sum 
of  the  arcs  intercepted  by  it  and  by  its  vertical. 
Given:    QC,  chords  AD  and  BE  intersecting  at  0. 
Prove:   $.AOB&  AB+DE 
Hint:  How  is  %.AOB  related  to  4»  A  and  E? 
SET  LXVII.    MEASUREMENT  OF  ANGLES 


668.  Fill  out  the  blank  spaces  in  the  table: 


3-1 

AE 

BD 

EB 

AND 

35° 

40° 

80° 

48° 

50° 

216° 

40° 

50° 

60° 

60° 

54° 

190° 

45° 

90° 

180° 

34° 

108° 

164° 

12 


178 


PLANE  GEOMETRY 


d669.   (a)  Prove  that  (I)  AB= 


,  etc.     (3) 
',  etc. 

(6)   Find  the   number  of  de- 
ID    grees  in  each  angle   mentioned 
in  (a). 

X  Theorem  54.  An  angle  whose 
vertex  is  outside  the  circle  is 
measured  by  half  the  difference 
of  its  intercepted  arcs. 


.  ^ 

AC -ED  in  Fig   lfAC-EA 


Given:  %.ABC  with  jertex  outside  GO,  inter- 
cepting in  Fig.  1  A£and  EJD;  in  Fig.  2  AC 
and  EA ',  in  Fig.  3^4 C  and  CA. 

.trove :  ^.^^^^  —  iAi  j.i&.  a., 

in  Fig.  2,  and  ARC~CA  in  Fig.  3. 

Hint:  In  Figs. 
>B  1  and  2  draw 
CX\\AB,and 
reduce  case 
3  to  case  2 
by  drawing 
any  secant 
BSR. 


^  Theorem 55.  A  tangent*  is  the  mean 
proportional  between  any  secant  and 
its  external  sect,  when  drawn  from 
the  same  point. 

Given:  0C,  AB  tangent  at  A  and  secant  BP 
cutting  GC  at  Q  and  P. 

VP  _AB  V 

^rr~  =  'rnzz  Jr 


Suggestion:  Prove 


*  By  "tangent"  in  such  cases  is  meant  the  sect  from  the  point  to  the  point 
of  tangency. 


THE  CIRCLE 


179 


EXERCISES.    SET  LXVIII.    TANGENT  AND  SECANT 

670.  (a)  Assuming  that  the  diameter  of  the  earth  is  8000  mi., 
how  far  can  a  man  see  from  the  top  of  a  building  200  ft.  high? 
(The  height  of  the  building  is  measured  along  the  prolongation  of 
the  diameter.) 

(6)  How  far  can  one  see  from  the  top  of  a  mountain  1000  ft. 
high?  2000ft.  high?  3000  ft.  high?  4000  ft.  high? 

(c)  How  far  can  one  see  from  a  balloon  1500  ft.  above  the  sea? 
d671.  Galileo  (1564-1642)  measured  the  heights  of  the  moun- 
tains on  the  moon,  some  of  which  are  as  much A      M      „ 

as  6  mi.  high,  as  follows:    ACS  was  the  illu- 
minated  half  of  the  moon  just  as  the  peak  of 
the  mountain  M  caught  the  beam  SM  of  the  c\ 
rising  or  setting  sun.     He  measured  the  dis- 
tance AM  from  the  half -moon's  straight  edge 
A B  to  the  mountain  peak  M.    Then  by  using 
the  known  diameter  of  the  moon,  show  how  he  was  able  to  com- 
pute the  height  of  the  mountain. 

d672.  Since  the  earth  is  smaller  than  the  sun,  it  casts  a  conical 
shadow  in  space  (umbra),  from  within  which  one  can  see  no  por- 
tion of  the  sun's  disk. 
If  S  is  the  center  of  the 
sun,  E  the  center  of  the 
earth,  and  V  the  end  or 
vertex  of  the  shadow, 
prove  that  the  length  of 

Approximately,    ES  =  92,900,000 


the  shadow,   VE= 


ESXEB 


SD  -EB 
miles,  SD  =  433,000  miles,  and  #£  =  4,000  miles.    Compute  VE. 

673.  Look  up  the  terms  "umbra"  and  "  penumbra,  "and  answer 
the  following  questions: 

(a)  How  are  the  umbra  and  penumbra  affected  by  a  change  in 
the  distance  apart  of  the  lumi- 
nous and  opaque  bodies? 

(6)  If  a  golf  ball  is  held  be- 
tween the  eye  and  the  sun,  is 
there  any  penumbra?  Explain. 


180 


PLANE  GEOMETRY 


(c)  What  may  be  said  of  the  umbra  if  the  luminous  body  and 

the  opaque  body  are  of  the  same  size? 
(d)  What  is  the  shape  of  the  umbra 
if  the  luminous  body  is  larger  than 
the  opaque  body,  as  in  the  case  of  the 
sun  and  the  earth? 

(Exs.  672-673  are  taken  from  Betz  and  Webb,  Plane  Geometry.) 

D.  MENSURATION  OF  THE  CIRCLE 

We  now  come  to  a  very  important  section  of  our  plane  geometry 
— the  section  which  develops  the  formula  for  the  length  of  the 
circle  and  for  the  area  enclosed  by  the  circle.  The  latter  is  known 
as  the  area  of  the  circle. 

A  polygon  is  said  to  be  cir- 
cumscribed about  a  circle  when     r      inscribed 
each  of  its  sides  is  tangent  to  the    fl         circle 
circle,  and  to  be  inscribed  in  a 
circle  when  each  of  its  sides  is 
a  chord  of  the  circle.    In  the 
first  case,  the  circle  is  said  to  be  inscribed  in  the  polygon,  and  in  the 
second  case,  the  circle  is  said  to  be  circumscribed  about  the  polygon. 

Theorem  56.  A  circle  may  be  circumscribed  about,  and  a  circle 
may  be  inscribed  in,  any  regular  polygon. 

Given:   The  regular  n-gon  PQRST  . . . 

To  prove:   1.  A  circle  may  be  circumscribed  about  PQRST  . . . 

2.  A  circle  may  be  inscribed  in  PQRST  .  .  . 
Hints  for  proof  1 : 

Three  non-collinear  points  determine 

a  circle. 
R  Draw  OP,  OQ,  OR,  OS. 

By  means  of  A  OPQ  and 

ORS  prove  OS=OP. 
To  what  triangle  can  AOST 

be  proved  congruent? 
Would  it  make  any  differ- 
ence how  many  sides  the 
original  polygon  had? 
Give  the  details  of  the 
proof. 


Let  0  be  the  center  of  the  circle 
determined  by  P,  Q,  and  R. 

B 


THE  CIRCLE 


181 


Hint  for  proof  2 :    If  0  is  the  center  of  the  circumscribed  circle,  what  are  PQ, 
QR,  RS,  etc.? 

Cor.  1.    An  equilateral  polygon  inscribed  in  a  circle  is  regular. 

Hint:  In  Fig.  1  what  is  the  measure  of  each  of  the  angles  P,  Q,  R,  etc.? 
Cor.  2.    An  equiangular  polygon  circumscribed  about  a  circle  is 
regular. 

Suggestion:  In  Fig.  2  connect  consecutive  points  of  contact  B,  C,  D,  etc. 
Prove  ABRC,  ACSD,  etc.,  congruent  isosceles  triangles. 

What  can  then  be  said  of  the  sums  of  any  two  of  the  equal  sides  of  the 
triangles?     (Such  as  RC+CS,  SD+DT,  etc.) 

An  angle  such  as  $.POQ  is  called  a  central  angle  of  the  regular 
polygon. 

EXERCISES.     SET   LXIX.       REGULAR  POLYGONS  AND   CIRCLES 

674.  Make  a  design  for  tiling  or  linoleum  pat- 
terns based  uponinscribed  equilateral  hexagt 

675.  Make  a  copy  of  the  accompanying 
design  of  arose  window  of  six  lobes.  (Fig.  1.) 

676.  Make    an   accurate    construction 
of  the  accompanying  design  representing 

a  Gothic  win- 
dow.   (Fig.  2.) 

d677.  Fig.  3 

shows  a  star  in-  Fig.  i. 

scribed  in  a  given  square  with  all  of  its 
vertices  on  the  sides  of  the  square. 

(a)  Construct  a  figure  in  which  points 
K,L,M,N,  etc.,  shall  be  the  mid-points 


B 


C- 

Fig.  2. 


D 


E 


of  AE,  EB,  BF,  FC,  etc. 

(6)  Show  that  a  circle  can  be  circum- 
scribed  about  the  star  constructed  as  in 
(a)  and  find  its  radius,  if  AB  =  a. 

678.  If  a  series  of  equal  chords  are  laid 
off  in  succession  on  a  circle,  what  relation 
exists  between: 

(a)  the  arcs  subtended  by    the  chords? 

(6)  the  central  angles  intercepting  the  arcs? 


Fig.  3. 


(c)  the  inscribed  angles  formed  by  any  two  successive  chords? 


182  PLANE  GEOMETRY 

679.  If  the  series  of  equal  chords  mentioned  in  the  last  exercise 
were  such  as  finally  to  form  an  inscribed  polygon,  what  kind  of 
polygon  would  it  be?    Why? 

680.  How  are  the  central  angles  of  a  regular  polygon  related? 

681.  Make  a  table  showing  the  number  of  degrees  in  the  central 
angle  of  a  regular  inscribed  polygon  of  3,  4, 20  sides. 

682.  Write  a  formula  by  means  of  which  we  can  obtain  the 
number  of  degrees  in  the  central  angle  of  any  regular  polygon. 

683.  Which  of  the  angles  found  for  the  table  in  Ex.  681  can 
we  construct  by  means  of  compasses  and  straight  edge? 

684.  Inscribe  in  a  given  circle  each  of  the  regular  polygons  you 
can  by  the  means  mentioned  in  the  last  exercise.     (Do  not  go 
beyond  the  sixteen-sided  polygon.) 

685.  Express  as  a  formula  the  number  of  sides  of  the  regular 
polygons  you  can  inscribe  in  a  circle  up  to  this  point. 

Theorem  57.  //  a  circle  is  divided  into  any  number  of  equal 
arcs,  the  chords  joining  the  successive  points  of  division  form  a 
regular  inscribed  polygon;  and  the  tangents  drawn  at  the  points  of 
division  form  a  regular  circumscribed  polygon. 

Given:  QO  with  AB  =  BC  = =  FA. 

(1)  Chords  AB,  BC FA. 

(2)  GH,  HK,  etc.,  tangent  to  QO  at  A,  B,  .... 
F  respectively. 

To  prove:  (1)  ABC . .  .  .F  a  regular  inscribed  polygon. 

(2)  GHK. .  .  .N  a  regular  circumscribed  polygon. 
Hints:  To  prove  (1)  use  Cor.  1,  Theorem  56. 

To  prove  (2)  use  Cor.  2,  Theorem  56. 

Cor.  1.  Tangents  to  a  circle  at  the  vertices  of  a  regular  inscribed 
polygon  form  a  regular  circumscribed  polygon  of  the 
same  number  of  sides. 

How  do  the  vertices  of  the  regular  inscribed  polygon  divide  the  circle? 

Cor.  2.  Lines  drawn  from  each  vertex  of  a  regular  polygon  to 
the  mid-points  of  the  adjacent  arcs  subtended  by  the 
sides  of  the  polygon  form  a  regular  inscribed  polygon 
of  double  the  number  of  sides. 

Hint :   Show  that  the  polygon  is  equilateral.    Why  ?     Or  throw  the  corol- 
lary back  to  the  proposition. 


THE  CIRCLE  183 

Cor.  3.  Tangents  at  the  mid-points  of  the  arcs  between  con- 
secutive points  of  contact  of  the  sides  of  a  regular 
circumscribed  polygon  form  a  regular  circumscribed 
polygon  of  double  the  number  of  sides. 

How  does  the  corollary  rest  on  the  theorem? 

Cor.  4.  The  perimeter  of  a  regular  inscribed  polygon  is  less 
than  that  of  a  regular  inscribed  polygon  of  double  the 
number  of  sides;  and  the  perimeter  of  a  regular  cir- 
cumscribed polygon  is  greater  than  that  of  a  regular 
circumscribed  polygon  of  double  the  number  of  sides. 
(Proof  left  to  the  student.) 

HISTORICAL  NOTE. — This  theorem  presupposes  the  possibility  of  dividing 
the  circle  into  any  number  of  equal  arcs.  In  the  table  found  in  answer  to 
Exercise ]681  it  was  probably  seen  that  the  number  of  equal  arcs  into  which 
we  are  at  this  time  able  to  divide  the  circle  is  very  limited,  but  we  have  not 
yet  learned  to  divide  the  circle  exactly  in  as  many  ways  as  possible  by  means 
of  elementary  geometry.  Some  other  methods  will  be  discussed  later. 

As  early  as  Euclid's  time  it  was  known  that  the  angular  magnitude  about 
a  point  (and  hence  a  circle)  could  be  divided  into  2n,  2n-3,  2«-5,  and  2n-15  ' 
equal  parts.  In  1796  it  was  discovered  by  Karl  Friedrich  Gauss,  then  only 
sixteen  years  of  age,  that  2«-17  equal  parts  of  the  circle  could  be  found  by  the 
use  of  only  the  straight  edge  and  compasses.  Gauss  also  showed  that  in 
general  it  is  possible  to  construct  all  regular  polygons  having  (2»+l)  sides, 
when  n  is  an  integer  and  (2n+l)  is  a  prime  number.  He  went  still  further  and 
proved  that  regular  polygons,  having  a  number  of  sides  equal  to  the  product 
of  two  or  more  different  numbers  of  this  series,  can  be  constructed. 

EXERCISES.    SET  LXIX  (continued) 

686.  Show  that  according  to  Gauss's  formula,  regular  polygons 
of  3,  5,  17,  and  257  sides  can  be  constructed. 

687.  Inscribe  a  square  in  a  circle,  and  by  means  of  it  a  regular 
octagon,  a  regular  16-sided,  and  a  regular  32-sided  polygon. 

688.  What  is  the  perimeter  of  the  square  in  terms  of  the  diameter 
of  the  circle  in  the  last  exercise?    How  do  the  perimeters  of  the 
octagon,  hexadecagon,  and  32-sided  polygons  compare  with  it? 
With  the  circle? 

689.  Repeat  exercises  687  and  688  with  respect  to  the  inscribed 
equilateral  triangle. 

690.  Repeat  exercises  687  and  680  with  respect  to  the  regular 
circumscribed  square  and  triangle. 


184  PLANE  GEOMETRY 

691.  Between  what  two  values  will  the  length  of  the  circle 
always  lie? 

692.  Between  what  two  values  will  the  area  of  the  circle 
always  lie? 

From  Cor.  4,  Theorem  57,  and  Exercises  688-692,  it  is  seen  that 
though  the  perimeter  of  the  inscribed  polygons  increases  as  the 
number  of  its  sides  increases,  it  is  always  less  than  the  length  of 
the  circle;  and  that  while  the  perimeter  of  the  circumscribed 
polygon  decreases  under  the  same  conditions,  it  is  always  greater 
than  the  length  of  the  circle. 

In  the  first  case  the  perimeter  and  likewise  the  area  of  the 
regular  inscribed  polygon  are  increasing  variables.  They  are 
always  less  than  the  perimeter  and  the  area  of  the  circle,  which 
are  fixed  or  constant. 

In  the  second  case,  the  perimeter  and  the  area  of  the  regular 
circumscribed  polygon  are  decreasing  variables  which  are  always 
greater  than  the  perimeter  and  the  area  of  the  circle. 

In  the  first  case  we  say  that  the  perimeter  and  the  area  approach 
as  superior  limits  the  circle  and  its  area,  while  in  the  second  we 
say  that  the  perimeter  and  the  area  approach  as  inferior  limits 
the  circle  and  its  area. 

Thus,  if  pn  represents  the  perimeter  of  a  regular  inscribed 
polygon  of  n  sides,  an)  its  area,  Pn  the  perimeter  of  a  regular  cir- 
cumscribed polygon  of  the  same  number  of  sides,  An,  its  area, 
and  c  the  circle  in  and  about  which  they  are  inscribed  and  cir- 
cumscribed, and  C  its  area,  we  say  that  as  the  number  n  H  in- 
creased, pn  approaches  c  as  its  limit,  Pn  approaches  c  as  its  limit, 
an  approaches  C  as  its  limit,  and  An  approaches  C  as  its  limit. 
These  statements  are  briefly  written  as  follows: 

pn=c}  Pn=c,  an  =  C,  An  =  C. 

POSTULATES  OF  LIMITS 

1.  The  circle  is  the  limit  which  the  perimeters  of  regular  inscribed 
and  circumscribed  polygons  approach  if  the  number  of  sides  of  the 
polygon  is  indefinitely  increased. 

2.  The  area  of  the  circle  is  the  limit  which  the  areas  of  regular 
inscribed  and  circumscribed  polygons  approach  as  the  number  of 
sides  of  the  polygon  is  increased. 


THE  CIRCLE 


185 


Theorem  68.    A  regular  polygon  the  number  of  whose  sides 
is  3-2"  may  be  inscribed  in  a  circle. 

(Proof  left  to  the  student.) 

EXERCISE.     SET  LXIX  (continued) 
693.  Inscribe  a  regular  polygon  of  3-24  sides. 

Theorem  59.    If  in  represent  the  side  of  a  regular  inscribed 

polygon  of  n  sides,  i2n  the  side  of  one  of  2n  sides,  and  r  the 

radius  of  the  circle,  i'2ft  =  V2r2  -  r\/4r*  -  ft. 

Given:  QO  of  radius  r,  AB  (in)  the  side  of  a  regular  in-  ~ 

scribed  n-gon,  AC  (i2n)  the  side  of  the  regular  inscribed          ^^ — -\ 
2n-gon.  j  ir^*  |      ^\g 

Prove:  i2n=  \/2r2-r\/4r2—in2. 

PROOF 

Draw  radius  OC  cutting  AB  at  D. 

Draw  AO- 

(1)  Then  OC  A.AB  and  AD=  %  (1)  'Why? 

(2)  .'.  AC2^(i2n)2^(  jjrl  +CD  (2)  Why? 

(3)  But  CD  =r-OD.  (3)  Why?    (Note  AOAD.) 


(5)   . ' .  t2n  =  \/2r  -  -  r  V±r2  -  inz 

EXERCISES.    SET  LXIX  (continued) 

694.  Given  a  circle  of  radius  1  unit,  compute: 
(a)  The  length  of  a  side  of  the  inscribed  square. 

(6)  The  length  of  a  side  of  the  inscribed  regular  octagon. 

(c)  The  length  of  a  side  of  the  inscribed  regular  16-sided  polygon. 

(d)  The  perimeters  of  each  of  these  polygons. 

695.  Do  the  same  as  you  did  in  the  last  exercise  for  the  regular 
hexagon,  dodecagon,  and  24-sided  polygon. 

696.  Do  the  same  as  you  did  in  Ex.  694,  using  a  diameter  of 
1  unit, 


186 


PLANE  GEOMETRY 


697.  Do  the  same  as  you  did  in  Ex.  695,  using  a  diameter  of 
1  unit. 

698.  (a)  Which  computation  is  simpler — that  using  a  radius 
of  1  unit  or  a  diameter  of  1  unit? 

(6)  If  the  radius  is  one  unit  what  must  be  done  to  the  perimeter 
found  for  each  of  the  polygons  in  order  that  it  be  expressed  in 
terms  of  the  diameter? 

Theorem  60.  If  in  represent  the  side  of  a  regular  inscribed 
polygon  of  n  sides,  cn  that  of  a  regular  circumscribed  polygon  of 

n  sides,  and  r  the  radius  of  the  circle,  cn  =    /^.^  2  • 

Given:  AB,  a  side  cn  of  a  regular  circum- 
scribed polygon  of  n  sides,  tangent  to  00 
of  radius  r  at  point  C;  in  the  side  of  a 
regular  inscribed  polygon  of  n  sides. 


A'* 


O 


'  Suggestions  for  proof:  Draw  AO,  BO,  CO. 
Let  AO  cut  QO  at  P,  and  BO  cut  it  at 

R,  and  CO  cut  PR  at  Q. 
Show  that  P/2=in. 

Note  that  OC  is  an  altitude  of  AAOB  and  OQ  of  APOfl. 
Why  is  AAOB  <*  APOfl? 

'  (1)  Why? 


(2)  In  AOPQ,  OQ  m  J  r'  -  f  | )      (2)  Why? 
Substitute  (2)  in  (1)  and  complete  the  proof. 

EXERCISE.     SET  LXIX  (concluded) 

699.  Repeat  the  computations  made  in  Exs.  694  and  695,  or  696 
and  697  for  the  sides  and  perimeters  of  regular  circumscribed 
polygons. 

The  radius  of  its  circumscribed  circle  is  called  the  radius  of  a 
regular  polygon,  and  the  radius  of  its  inscribed  circle  is  called  the 
apothem  of  a  regular  polygon. 

Cor.  The  apothem  of  a  regular  polygon  is  perpendicular  to 
its  side. 


THE  CIRCLE 


187 


Theorem  61.     The  perimeters  of  regular  polygons  of  the  same 
number  of  sides  compare  as  their  radii,  and  also  as  their  apothems. 

Given:  Polygons  ABCD 


Ni,  regular,  and  each 
of  n  sides,  and  with 
perimeters  p  and  p\t 
centers  0  and  Oi,  radii 
r  and  n,  and  apothems 
a  and  a\. 


To  prove:  (1)  -£-  s  -  and  (2)  2-  m  - 
Pi      ri  pi       a\ 

Suggestions:    Proof  of  (1) 
Show  that  AAOB  *> 
AB 


If 


21  why  is   P 

n  PI 


Proof  of  (2) 
What  are  a  and 


in  &AOB  and 


EXERCISE.     SET  LXX.      PERIMETERS  OF   REGULAR  POLYGONS 

700.  Construct  a  regular  hexagon  whose  perimeter  is  f  of  the 
perimeter  of  a  given  regular  hexagon. 

In  the  proofs  that  follow,  two  more  assumptions  are  made, 
namely : 

AXIOMS  OF  VARIABLES 

1.  //  any  variable  approaches  a  limit,  any  part  of  that  variable 
approaches  the  same  part  of  its  limit.    That  is,  if  v\  =  li,  then  /J=— 

2.  //  two  variables  are  always  equal,  the  limits  which  they  approach 
are  equal.    That  is,  if  v\=  l\,  and  v2  =  lz}  and  v\  =  v2  then  li  =  l*. 

Theorem  62.  Cir- 
cumferences have  the 
same  ratio  as  their 
radii. 

Given:  Circles  c  and  ea  of 
radii  r  and  rt. 

Prove:  e-JL. 


188 


PLANE  GEOMETRY 


PROOF 

Inscribe  in  each  circle  a  reg- 
ular polygon  of  n  sides,  and  let 
p  and  pi  be  their  perimeters. 

(1)  Then^-  s-. 
Pi      ri 


Let  n  increase  indefinitely. 
(3)  Thenp  =  candpi  =  ci  and 


(5) 


(1)  The  perimeters  of  regular  polygons  of  the 
same  number  of  sides  compare  as  their  radii. 

(2)  By  alternation. 

(3)  Postulate  of  limits  for  inscribed  poly- 
gons. 

(4)  If  any  variable  approaches  a  limit,  any 
part  of    that  variable    approaches  the  same 
part  of  its  limit  as  limit. 

(5)  If  two  variables  are  always  equal,  the 
limits  which  they  approach  are  equal. 

Cl    Tl  |      (6)  By  alternation. 

Cor.  1.    The  ratio  of  any  circle  to  its  diameter  is  constant. 

-=^-.     Why?  .'.   ~=£-.    Why? 

Ci    2ri  2r    2n 

Cor.  2.    Since  the  constant  ratio  ^  is  denoted  by  the  Greek 

letter  TT*  (which  is  the  initial  letter  of  the  Greek  word 
"periphery")  in  any  circle,  c=2irr. 

EXERCISE.    SET  LXXI.    VALUE  OF  it 

701.  Show  that  the  second  value  given  to  TT  by  Brahmagupta 
is  another  form  of  that  given  by  Ptolemy.  (See  historical  note.) 

Theorem  63.    The  value  of  TT  is  approximately  3.14159. 

If  pn  stands  for  the  perimeter  of  the  regular  inscribed  polygon, 
and  Pn  for  the  perimeter  of  the  regular  circumscribed  polygon  in 

*  HISTORICAL  NOTE. — "  Although  this  is  a  Greek  letter,  it  was  not  used 
by  the  Greeks  to  represent  this  ratio.  Indeed,  it  was  not  until  1706  that  an 
English  writer,  William  Jones,  used  it  in  this  way." 

Professor  D.  E.  Smith  further  tells  us  that  "probably  the  earliest  approxi- 
mation of  the  value  of  TT  was  3."  In  I  Kings,  vii,  23,  we  read :  "  And  he  made  a 
molten  sea,  ten  cubits  from  one  brim  to  the  other;  it  was  round  all  about — 


THE  CIRCLE  189 

a  circle  of  diameter  1,  the  following  table  can  be  derived  by  using 
the  formulas  of  the  last  two  theorems: 


(1)  i2n^2r2  -r\4r2  -in2  and  (2)  cn=    /JT         in  wbich  ^  i», 

V   *-T        ^",» 

in,  and  cn  retain  their  original  meanings. 

(The  student  might  verify  a  few  of  the  values  in  the  table  by 
using  logarithms  wherever  possible,  and  for  convenience  basing 
the  calculations  on  a  circle  of  diameter  2.) 

No.  of  sides.  pn     <     c     <     Pn 

6  3.0000000  3.4641016 

12  3.1058285  3.2153903 

24  3.1326286  3.1596599 

48  3.1393502  3.1460862 

96  3.1410319  3.1427146 

192  3.1414524  3.1418730 

384  3.1415576  3.1416627 

768  3.1415838  3.1416101 

1536  3.1415904  3.1415970 

EXERCISES.    SET  LXXII.     CIRCUMFERENCE 
g702.  Construct  a  graph  by  means  of  which  the  circumference 

of  a  circle  of  any  given  diameter  may  be  obtained. 

703.  Find  the  size  of  the  largest  square  timber  which  can  be 

cut  from  a  log  24  in.  in  diameter. 

and  a  line  of  thirty  cubits  did  compass  it  round  about."    Again  in  ii  Chronicles 

iv,  2,  we  read  a  similar  sentence.    And  again  in  the  Talmud  is  found  the  sen- 

tence: "What  is  three  hand-breadth  around  is  one  hand-breadth  through." 
The  following  list  of  various  other  values  given  to  TT  may  be  of  interest  to 

some  of  us. 

Value  Attributed  to 

3.106  .............................  Ahmes  (c.  1700  B.C.) 

Archimedes  (287-212  B.C.) 
Ptolemy  of  Alexandria  (87-165  A.D.) 

§§-8Jj-§  or  3.1416.  ...  .  ................  Aryabhatta  (c.  500  A.D.) 

f§§£  and  J!$  or  \/10  ...............  Brahmagupta  (c.  600  A.D.) 

m  ............................  Metius  of  Holland  (1571-1635) 

Computed  to  the  equivalent  of  over 

30  decimal  places  ..............  Ludolph  von  Ceulen  (1540-1610) 

To  140  decimal  places  ..............  Vega  (1756-1802) 

To  200  decimal  places  ..............  Base  (1824-1861) 

To  500  decimal  places  ..............  Richter  (1854) 

To  707  decimal  places  .............  Shanks  (1854) 


190 


PLANE  GEOMETRY 


G 


704.  The  following  problems  illustrate  the  type  of  problem  that 
is  suggested  by  books  on  carpentry. 

To  lay  off  an  octagon  on  the  end  of  a  square 
piece  of  timber  A  BCD,  draw  diagonals  AC  and 
BD.  With  radius  EF  (the  apothem  of  the 
square)  draw  arc  cutting  BD  at  G.  Square  out 
from  G.  Make  a  similar  construction  at  each  of 
the  other  corners  of  the  square.  Justify  the  rule. 
705.  How  much  belting  does  it  require  to 


A 

C 
D 

B 

E 

make  a  belt  to  run  over  two  pulleys,  each  30 
in.  in  diameter,  the  distance  between  their  centers  being  18  ft.? 

706.  The  annexed  figure  repre- 
sents a  small  wire  fence  used  to 
protect  flower  beds.  How  many 
feet  of  wire  are  needed  per  run- 
ning foot  of  fence  if  AB  =  1ft., 
BD  =  CD  =  9in.,  and  DE  =  3  in.? 
d707.  Using  4000  miles  as  the 
radius  of  the  earth,  find  the  num- 
ber of  feet  in  the  length  of  one  minute  at  the  equator.  Use  loga- 
rithms. 

(This  distance  is  commonly  called  a  "knot.") 
708.  (a)  If  a  cable  were  laid  around  the  earth  at  the  equator, 
how  many  feet  would  have  to  be  added  if  the  cable  were  raised 
10  ft.  above  the  surface  of  the  earth? 

(6)  If  the  same  were  done  around  a  gas-tank  whose  diameter 
is  100  ft.? 

(c)  In  which  case  is  the  increase  proportionally  larger? 

d709.  Carpenters  and  other  tradesmen 
frequently  wish  to  know  the  circumference 
of  a  circle  of  given  radius.  The  accompany- 
ing graphic  method  is  given  in  some  of  the 
self -education  books  as  a  substitute  for  com- 
putation : 

Draw  radii  AO  and  BO  at  right  angles. 
Draw  chord  AB  and  line  OE  perpendicular 
to  AB,  meeting  circle  at  E  and  chord  at  D. 


THE  CIRCLE 


191 


To  6  times  the  radius  add  the  sect  DE.  The  resulting  sect  is 
approximately  the  length  of  the  circumference. 

Compute  the  approximate  per  cent  of  error,  using  7r  =  3.14159. 

710.  The  central  angle  whose  arc  is  equal  to  the  radius  is  often 
used  as  the  unit  of  measure  of  angles.    It  is  called  a  radian.    Find 
the  number  of  degrees  in  a  radian. 

711.  Many  attempts  have  been  made 
to  construct  a  sect  equal  in  length  to  a 
circle.    The  following  approximate  con- 
struction is  one  of  the  simplest.     It  is 
due  to  Kochansky  (1685).  At  the  ex- 
tremity A  of  the  diameter  A  B  of  a  given     G 

circle  of  radius  r  draw  a  tangent  CD,  making  <£C£M=30°  and 
CD  =  3r.  Taking  BD  as  semi-circumference  is  equivalent  to  taking 
what  value  for  IT?  Carry  work  to  four  decimal  places. 

712.  A  running  track  consists  of  two  parallel  straight  stretches, 
each  a  quarter  of  a  mile  long,  and  two  semi-circular  ends,  each  a 
quarter  of  a  mile  long  at  the  inner  curb.     If  two  athletes  run, 
one  5  ft.  from  the  inner  curb  and  the  other  10  ft.  from  it,  by  how 
much  is  the  second  man  handicapped? 

713.  Show  how  to  go  into  the  field  and  lay  out  a  running  track 
of  the  dimensions  given  in  the  last  exercise. 

d714.  A  conduit  for  carrying  water  is 
circular  in  form  and  is  10'  in  diameter. 

(a)  Find  the  length  ABCD  of  the  por- 
tion of  the  circular  outline  which  is  wet 
when  the  water  reaches  AD  and  ^AED 
is  120°. 

(6)  Whatisthe  length  of  ABCD  if  <£AED 
is  60°? 

This  so-called  "wetted  perimeter"  is  of 
the  greatest  importance  in  determining 

friction,  and  therefore  the  resistance  of  the  pipe  to  the  water 
flow. 

d715.  Find  the  length  of  the  forty-second  parallel  of  latitude, 
assuming  the  radius  of  the  earth  to  be  4000  miles. 


192 


PLANE  GEOMETRY 


Theorem  64.  The  area  of  a  circle  is  equal 
to  one-half  the  product  of  its  radius  and  its 
circumference. 

Given:  O  0  of  radius  r,  area  C,  and  perimeter  c. 
Prove:  C  =  \cr. 


Write  out   the  proof,    giving  all 
authorities. 


PROOF 

Circumscribe  about  O#  a  regular 
polygon  of  n  sides  and  let  P  be  its 
perimeter  and  A  its  area. 

(1)  ShowthatA  =  £rP. 
Let  n  increase  indefinitely. 

(2)  Then  A  =  C  and  P  =c. 

(3)  .'.  |rP  =  |rc. 

(4)  .'.C^rc. 

Cor.  1.    The  area  of  a  circle  is  equal  to  n  times  the  square  of 
its  radius. 

(Substitution  left  to  the  student.) 

Cor.  2.    The  areas  of  circles  compare  as  the  squares  of  their 
radii. 

(Proof  left  to  the  student.) 

A  portion  of  the  area  of  a  circle  enclosed  by  two  radii  and  their 
intercepted  arc  is  a  sector. 

EXERCISES.    SET  LXXIII.    AREA  OF  CIRCLE 

g716.  Construct  a  graph  by  means  of  which  the  area  of  a  circle 
of  any  given  diameter  may  be  obtained. 

717.  Show  that  the  area  of  a  circle  is  equal  to  that  of  a  triangle, 
whose  base  equals  the  length  of  the  circumference  of  the  circle 
and  altitude  equals  the  radius.     This  was  proved  by  Archimedes. 

718.  Fill  in  the  blanks  in  the  following  table : 


Perimeter  Pi 

Area  Ai 

Perimeter  P2 

Area  A2 

Square 

300 

300 

Circle 

300 

300 

719.  Show  by  means  of  a  carpenter's  square  how  to  find  the 
diameter  of  a  circle  having  the  same  area  as  the  sum  of  the  areas 
of  two  given  circles. 


THE  CIRCLE 


193 


d720.  If  a  one-inch  pipe  will  empty  2  barrels  in  15  min.,  how 
many  barrels  will  an  8-in.  pipe  empty  in  24  hrs.?  (Make  no  allow- 
ance for  friction.) 

d721.  In  putting  up  blower  pipes,  two  cir- 
cular pipes  11  in.  and  14  in.  in  diameter  re- 
spectively join  and  continue  as  a  rectangular 
pipe  14  in.  in  width.  Find  the  length  of  the 
cross-section  of  the  rectangular  pipe. 

722.  A  convenient  formula  used  in  practical 
work  for  finding  the  area  of  a  "hollow  circle" 
or  ring  is :  irt(D+d) 

~2~ 
Establish  this  formula. 

723.  A  horse  tied  by  a  rope  25  ft.  long  at  the  corner  of  a  lot 
50  ft.  square,  grazes  over  as  much  of  the  lot  as  possible.    The  next 
day  he  is  tied  at  the  next  corner,  the  third  day  at  the  third  corner, 
and  the  fourth  day  at  the  fourth  corner.     Draw  a  plan  showing 
the  arcs  over  which  he  has  grazed  during  the  four  days,  using  a 
scale  of     inch  to  5  feet.    Calculate  the  area. 

724.  Justify  the  following  rule  used  by 
sheet-metal  workers  or  show  the  per  cent  of 
error  if  it  is  incorrect : 

Draw  radius  AO±OB.  Extend  each  one- 
fourth  its  own  length  to  C  and  D.  Then  the 
sect  CD  is  the  side  of  the 
square  of  the  same  area 
as  that  of  the  circle. 

d725.  Construct  a  square  with  a  side  s.    With 
the  vertices  as  centers,  and  s  as  radius,  construct 
arcs  as  in  the  figure.   Find 
the  perimeter  and  the  area 
of  the  shaded  portion  bounded  by  the  arcs. 

d726.  In  the  design  shown  in  this  figure, 
the  side  of  the  square  is  s.  The  inscribed 
semicircles  are  tangent  to  the  diagonals.  Find 
the  perimeter  and  the  area  of  the  shaded  por- 
tion of  the  figure. 
13 


194 


PLANE  GEOMETRY 


727.  Construct  an  equilateral  triangle.     With  each  vertex  as 
center,  and  with  one-half  a  side  as  radius,  describe  arcs  as  indicated 
in  the  diagrams.    Let  2r  represent  the  length 
0  of  a  side  of  the  equi- 

lateral triangle.  Find 
the  perimeter  and  the 
area  of  the  figure 
bounded  by  the  arcs. 

728.  Modify  the  prev 
ceding  exercise  by  using 
the  mid-point  of  the  sides  as  centers,  as  indi- 
cated in  the  diagrams.  /^Vv 
729.  Inscribe  an  equi-  [/  \\ 
lateral  triangle  in  a  circle  V  M 
of  radius  2r.  Using  the  /7  \\ 
mid-point  of  each  radius  I/  \\ 
of  the  triangle  as  center  ^^__^v^_^J> 
and  r  as  radius,  describe  circles.  Find  the  perimeter  and  the  area  of 
the  trefoil  and  of  the  shaded  part  of  the  resulting 
symmetric  pattern.  (Using  S  for  the  area  of 
the  shaded  portion,  C  for  that  of  the  large  circle, 
A  for  that  of  the  small  circle,  T  for  that  of  the 
triangle,  give  the  formula  for  S  in  terms  of  r, 


730.  In  the  papyrus  of  Ahmes,  an  Egyptian, 
the  area  of  a  circle  was  found  by  subtracting  from  the  diameter  one- 
ninth  of  its  length  and  squaring  the  remainder  This  was  equi- 
valent to  using  what  value  of  TT? 

731.  In  the  Sulvasutras,  early  semi-theological  writings  of  the 
Hindus,  it  is  said:  " Divide  the  diameter  into  15  parts  and  take 
away  2;  the  remainder  is  approximately  the  side  of  the  square, 
equal  to  the  circle."  From  this  compute  their  value  of  TT. 

d732.  The  proposition  of  the  so-called 
lunes  of  Hippocrates  (ca.  470  B.C.)  proved 
a  theorem  that  asserts  in  somewhat  more 
general  form,  that  if  three  semicircles  be 
described  on  the  sides  of  a  right  triangle  as 


THE  CIRCLE  195 

diameter,  the  lunes  L  and  LI  as  shown  in  the  diagram  are  together 
equivalent  to  the  triangle  T.     Prove  it. 

d733.  A  problem  of  interest  is  one  that  Napoleon  is  said  to  have 
suggested  to  his  staff  on  his  voyage  to  Egypt:  To  divide  a  circle 
into  four  equal  parts  by  the  use  of  circles  alone. 

LIST  OF  WORDS  DEFINED  IN  CHAPTER  VDI 

Circle,  center,  radius,  diameter;  central  angle,  arc  (minor,  major);  chord, 
intercept,  subtend;  tangent  to  a  circle,  secant,  tangent  circles  (internally, 
externally);  line  of  centers,- common  chord,  inscribed  angle;  inscribed,  circum- 
scribed regular  polygons;  center,  central  angles,  radius,  apothem  of  regular 
polygon;  sector.  Constant,  variables  (increasing,  decreasing);  limits  (inferior, 
superior). 

AXIOMS  OF  VARIABLES  IN  CHAPTER  Vm 

1.  If  any  variable  approaches  a  limit,  any  part  of  that  variable  approaches 
the  same  part  of  its  limit,  as  limit. 

2.  If  two  variables  are  always  equal,  the  limits  which  they  approach  are  equal. 

POSTULATE  OF  PERPENDICULARS  (third)  IN  CHAPTER  VHI 

1.  The  shortest  distance  from  a  point  to  a  line  is  the  perpendicular  to  that 
line. 

SUMMARY  OF  THEOREMS  PROVED  IN  CHAPTER  Vm 

42.  Three  points  not  in  a  straight  line  fix  a  circle. 

43.  In   equal   circles   equal    central    angles    intercept   equal   arcs,   and 
conversely. 

44.  In  equal  circles,  equal  arcs  are  subtended  by  equal  chords,  and 
conversely. 

45.  A  diameter  perpendicular  to  a  chord  bisects  it  and  its  subtended  arcs. 

Cor.  1.    A  radius  which  bisects  a  chord  is  perpendicular  to  it 
Cor.  2.    The  perpendicular  bisector  of  a  chord  passes  through  the 
center. 

46.  In  equal  circles,  equal  chords  are  equidistant  from  the  center,  and 
conversely. 

47.  A  line  perpendicular  to  a  radius  at  its  outer  extremity  is  tangent  to 
the  circle. 

Cor.  1.    A  tangent  to  a  circle  is  perpendicular  to  a  radius  drawn 

to  the  point  of  contact. 
Cor.  2.    The  perpendicular  to  a  tangent    at   the   point  of   contact 

passes  through  the  center  of  the  circle. 
Cor.  3.    A  radius  perpendicular  to  a  tangent  passes  through  the  point 

of  contact. 
Cor.  4.    Only  one  tangent  can  be  drawn  to  a  circle  at  a  given  point 

on  it. 


196  PLANE  GEOMETRY 

48.  Sects  of  tangents  from  the  same  point  to  a  circle  are  equal. 

49.  The  line  of  centers  of  two  tangent  circles  passes  through  their  point 
of  contact. 

50.  In  equal  circles  central  angles  have  the  same  ratio  as  their  intercepted 
arcs. 

Cor.  1.    A  central  angle  is  measured  by  its  intercepted  arc. 

51.  Parallels  intercept  equal  arcs  on  a  circle. 

52.  An  inscribed  angle  or  one  formed  by  a  tangent  and  a  chord  is  measured 
by  one-half  its  intercepted  arc. 

Cor.  1.    An  angle  inscribed  in  a  semicircle  is  a  right  angle. 

53.  An  angle  whose  vertex  is  inside  the  circle  is  measured  by  half  the  sum 
of  the  arc  intercepted  by  it  and  by  its  vertical  angle. 

54.  An  angle  whose  vertex  is  outside  the  circle  is  measured  by  half  the 
difference  of  the  intercepted  arcs. 

55.  A  tangent  is  the  mean  proportional  between  any  secant  and  its  external 
sect,  when  drawn  from  the  same  point. 

56.  A  circle  may  be  circumscribed  about,  and  a  circle  may  be  inscribed  in, 
any  regular  polygon. 

Cor.  1.    An  equilateral  polygon  inscribed  in  a  circle  is  regular. 
Cor.  2.    An  equiangular    polygon  circumscribed  about  a    circle   is 
regular. 

57.  If  a  circle  is  divided  into  any  number  of  equal  arcs,  the  chords  joining 
the  successive  points  of  division  form  a  regular  inscribed  polygon,  and  the  tan- 
gents drawn  at  the  points  of  division  form  a  regular  circumscribed  polygon. 

Cor.  1.  Tangents  to  a  circle  at  the  vertices  of  a  regular  inscribed 
polygon  form  a  regular  circumscribed  polygon  of  the 
same  number  of  sides. 

Cor.  2.  Lines  drawn  from  each  vertex  of  a  regular  polygon  to  the 
mid-points  of  the  adjacent  arcs  subtended  by  the  sides  of 
the  polygon  form  a  regular  inscribed  polygon  of  double 
the  number  of  sides. 

Cor.  3.  Tangents  at  the  mid-points  of  the  arcs  between  consecutive 
points  of  contact  of  the  sides  of  a  regular  circumscribed 
polygon  form  a  regular  circumscribed  polygon  of  double 
the  number  of  sides. 

Cor.  4.  The  perimeter  of  a  regular  inscribed  polygon  is  less  than 
that  of  a  regular  inscribed  polygon  of  double  the  num- 
ber of  sides,  and  the  perimeter  of  a  regular  circumscribed 
polygon  is  greater  than  that  of  a  regular  circumscribed 
polygon  of  double  the  number  of  sides. 

58.  A  regular  polygon  the  number  of  whose  sides  is  3*2W  may  be  inscribed 
in  a  circle. 

59.  If  in  represent  the  side  of  a  regular  inscribed  polygon  of  n  sides,  and  i2re 


the  side  of  one  of  2n  sides,  and  r  the  radius  of  the  circle,  i2n  =  V2r2  —  r\/4r2—  in 


THE  CIRCLE 


197 


60.  If  ira  represent  the  side  of  a  regular  inscribed  polygon  of  n  sides,  cn 
that  of  a  regular  circumscribed  polygon  of  n  sides,  and  r  the  radius  of  the  circle, 

2rira 

C7l  =  VV-ire2' 

61 .  The  perimeters  of  regular  polygons  of  the  same  number  of  sides  com- 
pare as  their  radii,  and  also  as  their  apothems. 

62.  Circumferences  have  the  same  ratio  as  their  radii. 

Cor.  1.    The  ratio  of  any  circle  to  its  diameter  is  constant. 
Cor.  2.    In  any  circle  c =2-jrr. 

63.  The  value  of  IT  is  approximately  3.14159. 

64.  The  area  of  a  circle  is  equal  to  half  the  product  of  its  radius  and  its 
circumference. 

Cor.  1.    The  area  of  a  circle  is  equal  to  IT  times  the  square  of  its 

radius. 
Cor.  2.    The  areas  of  circles  compare  as  the  squares  of  their  radii. 


21  ft. 


EXERCISES.     SET  LXXIV.  MISCELLANEOUS 
The  problems  in  this  set  are  miscellaneous  in  that  their  solution 
may  depend  upon  any  part  of  the  text;  but  they  are  arranged,  in 
general,  in  the  order  of  difficulty.    Those  problems  requiring  the 
application  of  trigonometric  ratios  are  preceded  by  "t." 

734.  Find  the  num-      73  ft. 

ber  of  feet  of  lime-line  1 4-5  ft> 

of  a   tennis-court,  as 

represented. 

735.  Find  the  num- 
ber of  yards  of  lime- 
line  for  a  football  field, 
which  is  300  ft.  by  160 

ft.,  including  all  the  ten-yard  lines.    How  long  would  it  take  a  run- 
ner to  cover  the  total  distance  if  he  can  make  40  feet  in  12  seconds? 

736.  Construct  an  accurate  diagram  of  a  rectangular  garden 
with  a  border  inside  it  one-fourth  the  width  of  the  garden. 

737.  A  designer,  in  making 
a  pennant,  must  make  one  in 

B  the  same  proportion  as  a  given 
one,  but  larger.  Find  AB,  if 

D  CD  =  36",   A&  =  43",    and 


198 


PLANE  GEOMETRY 


E 


738.  By  use  of  the  steel  .square  lay  out  an  angle  of  45°. 

739.  By  use  of  steel  squares  lay  out  an  angle  of  60°. 

740.  A  straight  railroad  AB  strikes  a  mountain  at  C,  and  a 

tunnel  is  to  be  driven  at  C  in 
the  direction  ABC.  It  is  de- 
sired to  commence  the  work  at 
the  other  side  of  the  mountain 
at  the  same  time  as  the  work 
is  progressing  from  C.  -&A BD 
ismade  150°,  BD  =  3 miles,  and 
60°.  How  far  from  D  in 


DE  must  the  tunnel  be  driven,  and  in  what  direction? 
741.    An  instrument  for  leveling  A  ^ 

consists  of  a  rectangular  frame  ABCD. 

E  and  F  are  the  mid-points  of  A B  and 

DC,  respectively.      A  plumb-line  is 

suspended  from  E.    Show  that  when 

the    plumb-line  coincides   with   the 

mark  F}  DC  is  horizontal. 

This  instrument  is  shown  in  French  books. 

B  742.  The   accompanying 

diagrams  represent  an  in- 
strument for  locating  the 
center  of  circular  discs. 
Three  pieces  of  metal  are  so 
joined  that  AB  bisects  the 
angle  formed  by  BD  and  BE, 
which  two  sects  are  equal. 

Prove  that  AB  passes  through  the  center  of  the  circle. 

743.  A  carpenter  bisects  an  angle  by  the 
following  rule:  Lay  off  AB=AC.    Place 
a  steel  square  so  that  BD  =  CD  as  shown 
in  the  diagram.    Draw  the  line  AD.    Is 
this  method  correct?  Give  proof.   Would 
this  method  be  correct  if  the  steel  instru- 
ment did  not  have  a  right  angle  at  D? 

744.  Answer  the  following  questions  without  proof: 


THE  CIRCLE  199 

(a)  Are  all  equilateral  polygons  equiangular? 
(6)  Are  the  diagonals  of  a  parallelogram  equal? 

(c)  A  diagonal  of  a  parallelogram  divides  the  figure  into  two 
congruent  triangles.    Is  this  proposition  conversely  true? 

(d)  Do  the  diagonals  of  a  parallelogram  bisect  its  angles? 

(e)  Under  what  circumstances  do  two  chords  bisect  each  other? 
(/)   An  arc  of  30°  is  subtended  by  a  chord  of  5".    In  the  same 

circle  will  an  arc  of  60°  be  subtended  by  a  chord  equal  to,  .greater 
than,  or  less  than  5"? 

(g)  Is  it  possible  to  inscribe  a  parallelogram  in  a  circle? 

745.  In  an  ideal  honeycomb  the  cells  are  6  sided.    Why  is  this? 
In  what  other  regular  form  might  they  be  built  and  yet  fit  snugly? 

746.  The  wheel  of  an  automobile  makes  110  revolutions  per 
minute.    If  it  measures  2  ft.  6  in.  in  diameter,  find  the  speed  of  the 
machine. 

747.  Walking  along  a  straight  road  a  traveler  noticed  at  one 
milestone  that  a  house  was  30°  off  to  the  right.    At  the  next  mile- 
stone the  house  was  45°  off  to  the  right.    How  far  was  the  house 
from  the  road?    Is  there  more  than  one  solution? 

748.  Calculate  the  diameter  of  the  circle  of  water  visible  to  an 
observer  at  sea,  (a)  when  seated  in  a  small  boat,  his  eyes  being  4  ft. 
above  the  surface  of  the  sea,  (b)  when  on  the  bridge  of  a  steamer 
25  ft.  above  the  surface,  (c)  when  at  the  masthead  60  ft.  above  the 
surface,  (d)  when  on  the  top  of  a  mountain  3000  ft.  above  sea  level. 
(e)  How  far  above  sea  level  does  the  elevation  of  the  observer 
begin  to  make  a  perceptible  difference? 

a749.  In  finding  the  diameter  of  a  wrought-iron  shaft  that  will 
transmit  90  horse-power  when  the  number  of  revolutions  is  100 
per  minute,  using  a  factor  of  safety  of  8,  we  have  to  find  the 

3/          90 
diameter  d  from  the  formula  d=68.5  A  /  50000.    Find  d. 


o 

750.  Draw  any  quadrilateral  ABCD.  Take  such  measurements 
of  your  figure  as  you  consider  necessary  and  sufficient,  and  from 
your  measurements  construct  the  quadrilateral  a  second  time. 
State  what  measurements  you  make,  and  how  you  draw  the  second 
figure.  Cut  out  the  two  figures  and  fit  one  upon  the  other. 


200 


PLANE  GEOMETRY 


(6)  Discuss  a  number  of  other  sets  of  measurements  that  you 
could  use  to  reproduce  the  quadrilateral  A  BCD.  How  many  meas- 
urements must  there  be  in  every  set? 

(c)  Construct  a  quadrilateral 
similar  to  ABC 'D,  but  50%  greater 
in  area. 

751.  Find  the  number  of  square 
feet  in  the  floor  of  the  room  shown 
in  the  accompanying  plan. 

752.  Find  the  area  of  the  ac- 
companying polygon  by  filling  out 

the  following  table,   assuming  reasonable  values  for  necessary 
dimensions.     (Fig.  1). 


Parts  of 
Polygon 

Factors 

Products 

Bases,  or  sums  of 
parallel  sides 

Altitudes 

AABB! 

BBi  =  6.8 

4£i=5-G 

38.08 

2)   ' 

Polygon  ABCDEFGH  = 

B 


753.  Show  how  to  find  the  area  of 
polygon  ABCDEFGHK,  assuming  the 
shaded  portion  to  be  inaccessible. 


rs. 


FIG.  2. 


754.  Using  goods  20  in.  wide,  how  many  strips  will  it  take, 
cut  on  true  bias,  to  put  a  band  12  in.  wide  around  a  skirt  3  yds. 
wide? 


THE  CIRCLE  201 

755.  In  the  accompanying  diagram  how  do  (a)  the  perimeters 
and  (6)  the  areas  of  the  circle  and  the  curvi- 
linear figures  ADCFBX  compare?      (c)  Use 
this  figure  as  a  suggestion  and  show  how, 
by  means  of  arcs,  to  divide  a  circle  into  any 
number  of  equal  areas. 
756.  This  is  the  cross- 
section  of  a  foot-stool, 
in  which  the  width  of 
the  top  is  to  be  12  in.,  d  8  in.,  e  12  in.,  and 
the  lengths  of  the  legs  8  in.    In  making  the 
stool,  angle  a.  and  angle  /?  are  first  laid  out 
on  paper.    Show  how,  from  the  required  di- 
mensions, to  lay  out  these  angles  on  paper. 

757.  To  find  the  diagonal  of  a  square,  multiply  the  side  by  10, 
take  away  1%  of  this  product,  and  divide  the  remainder  by  7. 
Test  the  accuracy  of  this  rule  of  thumb  used  by  some  carpenters. 

758.  Construct   a   perpendicular  at  the 
end  of  a  sect  without  producing  the  sect. 

Hints:  Let  AB  be  the  given  sect.  With  any 
point  C,  between  A  and  B,  but  outside  the  sect,  as 
center,  and  radius  CB,  describe  a  major  arc  inter- 
secting AB  at  E.  Draw  the  diameter  EK.  KB  is 
the  required  perpendicular. 

Prove  that  this  construction  is  correct. 

759.  The  resultant  of  two  forces  acting  upon  a  body  is  400  Ibs. 
One  of  the  forces  is  250  Ibs.    What  are  the  limiting  values  for  the 
other  force? 

760.  A  man  decided  to  buy  some  numerals  and  make  the  face 
of  a  grandfather's  clock,  but  when  he  came  to  divide  the  face  into 
minutes  he  found  that  he  was  not  able  to  do  it  without  gues- 
sing.   How  could  it  be  done  accurately? 

t761.  A  kite  string  is  250  ft.  long,  and  makes  an  angle  of  40° 
with  the  level  ground.  Find  (approximately)  the  height  of  the 
kite  above  the  ground,  disregarding  the  sag  in  the  string. 


202 


PLANE  GEOMETRY 


762.  Fill  out  the  blank  spaces  in  the  table  by  referring  to  the 
diagram.     (Fig.  1.) 


43 

41 

3-4 

4-5 

EA 

DB 

4NZ> 

AF 

BE 

^1G 

FH 

41 

35° 

41 

40° 

80° 

30° 

2L3 

4-3 

30° 

50° 

190° 

105° 

FIG.  2 


763.  The  accompanying  drawing  is  one 

of  the  earliest  of  Gothic    O 

tracery  windows.    The 

arch^.C#  is  based  on 

an  equilateral  trian- 
gle. A  is  the  center 

and  AB  theradius 

for  the  arc  CB, 

and.B  the  cen-  A\ 

ter     and    AB 

the   radius  for 

arc  AC.  D  is  the  mid-point  of  span  AB.  The  arches  AED 
and  DFB  are  drawn  on  the  half  span  by  similar  construction. 
Find  the  center  and  the  radius  of  the  circle  with  center  0  that 
shall  be  tangent  to  the  four  arcs,  DE,  DF,  AC,  CB. 

Suggestions:    With  A  as  center  and  AH  as  radius  cut  the  altitude  CD  at  0- 
H  is  the  mid-point  of  DB. 

764.  How  many  degrees  are  there  in  each  of  the  angles  of  the 
Pythagorean  badge? 

765.  From  a  strip  of  metal  2^"  wide 
it  is  desired  to  cut  off  a  rectangle  from 
which  two  circular  disks  2"  in  diameter 
can  be  cut.    What  length  AB  must  be 
cut  off? 

766.  In  taking  soundings   to    make  a  chart  of  a  harbor  it  is 
necessary  at  each  sounding  to  determine  the  position  of  the  boat. 
This  is  sometimes  done   by  measuring  with  a  sextant  the  angles 
between  lines  from  the  observer  to  three  range  poles  on  the  shore. 
When  a  chart  is  made,  the  position  on  the  chart  of  each  sounding 
is  sometimes  found  as  follows:     The  points  A,  B,  and  C  represent 
the  positions  of  the  three  range  poles.      Suppose  the  angles  read 


THE  CIRCLE  203 

by  the  sextant  were  50°,  and  35°  (50°  between  lines  to  A  and  B, 
and  35°  between  lines  to  B  and  C) .  On  the  chart  lay  off  at  A  and 
B,  angles  BAM  and  ABM,  each  equal  to  40°,  and  find  the  point 
M.  At  B  and  C,  lay  off  angles  CBN  and  BCN  each  equal  to  55° 
and  find  N.  With  M  and  N  as  centers,  draw  circles  passing  through 
B.  The  other  common  point  of  these  circles  is  the  position  of  the 
sounding.  Prove  the  correctness  of  this  construction.  Make  the 
construction  to  scale. 

767.  The  resistance  offered  by  the  air  to  the  passage  of  a  bullet 
through  it  varies  jointly  as  the  square  of  its  diameter  and  the 
square  of  its  velocity.    If  the  resistance  to  a  bullet  whose  diameter 
is  .32  in.,  and  whose  velocity  is  1562.5  ft.  per  second,  is  67.5  oz., 
what  wiJl  be  the  resistance  to  a  bullet  whose  diameter  is  .5  in.,  and 
whose  velocity  is  1300  ft.  per  second? 

768.  Roman  surveyors,  called  agrimensores,  are  said  to  have 
used  the  following  method  of  measuring  the  width  of  a  stream: 
A  and  B  were  points  on  opposite  sides  of  the  stream,  in  plain 
view  from  each  other.    The  distance  AD  was  then  taken  at  right 
angles  to  AB,  and  bisected  at  E.    Then  the  distance  DF  was  taken 
at  right  angles  to  AD,  such  that  the  points  B,  E,  and  F  were  in  a 
straight  line.    Make  a  drawing  illustrating  the  above  method,  show 
what  measurement  affords  a  solution,  and  prove  that  this  is  so. 

769.  A  student  lamp  and  a  gas  jet  illuminate  a  screen  equally 
when  it  is  placed  12  ft.  from  the  former  and  20ft.  from  the  latter. 
Compare  the  relative  intensities  of  the  two  lights. 

770.  An  endless  knife  runs  on  pulleys  48"  in  diameter  at  the 
rate  of  180  revolutions  per  minute.    If  the  pulleys  are  decreased 
18"  in  diameter,  how  many  revolutions  per  minute  will  they  have 
to  make  to  keep  the  knife  traveling  at  the  same  speed? 

771.  In  surveying,  to  determine 
a  line  from  the  inaccessible  point  P 
perpendicular  to  AB,  lay  off  FE 
perpendicular  to  AB  at  an  arbit- 
rary    point    and   of    any   length. 
Make  EH  =  EF.    Obtain  point  D 

in  lines  PF  andAB;  next,  point  N  in  HP  and  AB;  next,  K  in  DH 
and  FN.  PK  is  perpendicular  to  AB.  Why? 


204  PLANE  GEOMETRY 

772.  Make  the  construction  shown  in  the  diagram,  and  AB  will 
be  approximately  the  quadrant  of  the 
circle.  Find  the  per  cent  by  which  it 
differs  from  the  correct  value. 

773.  The  steel  square  may  be  tested  by 
measuring  across  from  the  9-inch  point 
of  the  tongue  to  the  12-inch  point  of  the 
blade.     If  this   distance  is  exactly   15 
inches,  the  square  is  true.    Why? 
774.  Four  of  the  largest  possible  equal  sized  pipes  are  enclosed 
in  a  box  of  square  cross-section  18  in.  on  an  edge.    What  part  of 
the  space  do  the  pipes  occupy? 

dt775.  A  boy  pulls  a  sled  with  a  force  of  twenty-five  pounds  by 
means  of  a  rope  ten  feet  long,  and  with  his  hands  three  feet  from  the 
ground.  Find  the  component  of  his  force  effective  in  pulling  the 
sled  forward. 

776.  A  camp  kettle  weighing    20  Ibs.  is  suspended  on  a  wire 
from  two  trees  10  feet  apart.    The  wire  is  20  feet  long,  and  the 
kettle  is  suspended  on  it  at  a  point  midway  between  the  trees. 
Find  the  tension  on  each  strand  of  the  wire. 

777.  A  running  track  having  two  parallel  sides  and  two  semi- 
circular ends,  each  equal  to  one  of  the  parallel  sides,  measures 
exactly  a  mile  at  the  inner  curb.    Two  athletes  run,  one  at  the 
inner  curb,  and  the  other  10  ft.  from  this  curb.    By  how  much  is 
the  second  man  handicapped? 

778.  The  last  row  of  seats  in  a  circular  tent  is  30  ft.  away  from 
the  central  pole,  which  is  20  feet  high,  and  which  is  to  be  fastened 
by  ropes  from  its  top  to  stakes  driven  in  the  ground.    How  long 
must  these  ropes  be  in  order  that  they  may  be  6  feet  above  the 
ground  over  the  last  row  of  seats,  and  at  what  distance  from  the 
center  must  the  stakes  be  driven? 

779.  A  wheelwright  is  given  a  part  of  a 
broken  wheel  to  make  a  duplicate.    To  do 
this  he   needs   the  diameter.     He  measures 
the  chord  of  the  arc  given  him,  24";  the 
height  of  the  segment  is  4".    How  large  is 
the  wheel? 


THE  CIRCLE 


205 


dt780.  The  width  of  the  gable  of  a  house 
is  35  ft.    The  height  of  the  house  above  the  , 
eaves  is  15  ft.    Find  the  length  of  the  rafters 
and  the  angle  of  inclination  of  the  roof. 

781.  An  aeroplane  travels  1000  ft.  upwards,  3^  miles  due  west, 
and  2-^  miles  due  north.    Find  its  distance  from  the  starting-point. 

782.  The  sides  of  a  floor  are  10  ft.  and  6  ft.    A  man  wishes  to 
tile  it  with  tiles  in  the  shape  of  a  hexagon  whose  sides  are  6  in. 
How  many  tiles  will  it  take? 

783.  A  flat  circular  sheet  of  metal   is  to  be  stamped  into  the 
form  of  a  spherical  segment  with  a  flange.     The  figure  shows  a 
cross-section  of  the  resulting  piece  of  metal,  a  being  the  width  of  the 

spherical  segment,  b  the  depth  or  altitude 
of  the  segment,  and  c  the  outer  diameter 
of  the  flange.  The  problem  is  to  determine 
the  size  to  cut  the  sheet  metal  in  order 
that  when  stamped  the  piece  may  have  these  dimensions.  Show 
that  the  required  radius  of  the  circular  sheet  equals  e  in  the  figure. 

784.  To  lay  off  the  length  of  a  brace  with  the  steel  square: 
Suppose  that  the  post  is  4  ft.  and  the  beam  3  ft.     Apply  3  times 
to  the  timber  from  which  the  brace  is  to  be  cut,  the  distance  across 
the  square  from  the  12  in.  point  of  the  tongue  to  the  16  in.  point 
of  the  blade.    Why  will  this  give  the  required  length  of  the  brace? 

t785.  In  railroad  construction  and  mining  the  material  is 
sometimes  hauled  in  a  tram  pulled  by  a  horse.  If  the  pull  of  the 
tram  in  the  direction  of  the  track  is,  say,  200  Ibs.,  and  if  the  horse 
walks  at  the  side  of  track  so  that  its  pull  is  exerted  at  an  angle  of 
25°  with  the  track,  what  pull  must  the  horse  exert? 

786.  The  force  that  the  wind  exerts  normally  (per- 
pendicularly) on  the  sail  of  a  boat  is  resolved  into 
two  components:  one  useless,  in  pushing 
the  boat  sidewise  in  the  water  in 
spite  of  the  keels;  and  one  useful  com- 
ponent driving  the  boat  directly  forward.  If,  as  in  the  following 
diagram,  the  sail  is  at  an  angle  of  30°  to  the  keel,  and  a  force  of 
wind  of  100  Ibs.  acts  on  the  sail  (considered  as  applied  at  one  point), 
find  the  effective  value  of  the  two  components  mentioned  above. 

NOTE. — A  similar  problem  can  be  applied  to  the  aeroplane. 


206 


PLANE  GEOMETRY 


787.  An  angle  inscribed  in  a  circle  varies  directly  as  the  inter- 
cepted arc.    Show  that  in  this  case  k  =  ^. 

788.  A  street-car  track  is  12'  from  the  curb  (GF  =  BC=12f). 

In  passing  the  corner  of  two  streets 
which  deflect  through  an  angle  of 
60°  the  rail  must  be  5'  (DE  =  &) 
from  the  corner,  (a)  Find  the 
radius  of  the  curve.  (6)  Find  the 
length  of  the  tangents  from  G  and 
C  to  their  point  of  intersection. 
(c)  Find  the  length  of  arc  GC;  also 
the  length  of  the  outer  arc.  The 
width  of  the  track  is  4'  &|-".  (All  these  curves  are  arcs  of  circles.) 
789-  Where  two  straight  streets  intersect,  each  corner  is  usually 
"rounded  off."  Show  that  the  problem  of  laying  out  the  corner 
arc  is  a  simple  one  where  the  streets  made  an  angle  of  90°.  Show 
by  a  plan  how  to  lay  out  a  corner  where  the  angle  is  45°,  and  the 
radius  for  the  curbing  is  to  be  20  ft.  Find  the  total  length  of 
curved  curbing  needed. 

dt790.  A  given  regular  polygon  has  n  sides.  How  many  meas- 
urements of  the  figure  are  both  necessary  and  sufficient  to  deter- 
mine it  in  size  and  shape? 

dt791.  A  flagstaff  is  seen  in  a  direction  due  north  of  a  station 
A  at  an  elevation  of  17°,  and  from  a  station  B  120  ft.-  due  east  of 
A  the  flagstaff  bears  23°  west  of  north.  The  two  stations  and  the 
foot  of  the  flagstaff  being  at  the  same  level,  determine  the  height 
of  the  flagstaff. 

792.  A  straight  street  intersects 
one  which  is  curved  (with  a  large 
radius,  such  as  200  ft.),  and  the 
corner  is  to  have  a  small  radius, 
say  15  ft.    Show  in  a  plan  how  to 
find  the  center  for  the  corner  arc 

by  means  of  the  intersection  of  two  loci.     (Fig.  1.) 

How  would  you  get,  in  actual  field  work,  the  curved  locus? 
(Note  its  large  radius.)  (Fig.  2.) 

793.  Solve  in  another  plan  the  corresponding  problem  where 
both  streets  are  curved. 


Fig.  1. 


Fig.  2. 


THE  CIRCLE 


207 


4'lK"- 


D 


794.  The  "gear"  of  a  bicycle  is  the  diameter  (in  inches)  of  a 
wheel  whose  circumference  would  equal  the  distance  gone  forward 
with  one  revolution  of  the  pedals.    Find  the  gear  if  the  diameter 
of  the  rear  wheel  is  28"  and  the  front  and  rear  sprockets  have  22 
and  8  teeth,  respectively. 

795.  A  belt  runs  over  two  pulleys,  one  of  which  is  4  ft.  in  dia- 
meter, and  driven  by  an  engine  at  the  rate  of  100  revolutions  a 
minute.    What  must  be  the  diameter  of  the  other  pulley  if  it  is 
to  turn  a  fan  at  the  rate  of  400  revolutions  a  minute? 

796.  The  figure  is  the  diagram  of  a  part 
of  the  side  of  a  bridge.   The  point  C  must 
be  located  on  AB  and  on  DEy  where  the 
holes  must  be  bored  to  fasten  the  brace 
AB  to  the  upright  DE.    Required  to  find 
the  lengths  of  AB,  x,  y,  and  2,  in  order  to 
locate  C. 

Suggestion:     After    finding    AB,     compare 
triangles  ACF  and  ABG. 

797.  It  is  desired  to  construct  a  subway 
under  a  river.    The  bank  on  one  side  has 
a  30%  slope  from  the  edge  of  the  river  to 
the  river  bed.     The  maximum  effective 
grade  of  a  subway  is  5%.   On  the  surface 
it  is  five  hundred  feet  from  the  bank  of 
the  river  to  the  bed  of  the  river.    Deter- 


E 


K-    2  !&--;*: 2' "I 


mine  the  necessary  length  of  the  subway 
from  the  bank  to  the  river  bed. 

798.  WW  is  a  wall  with  a  round  cor- 
ner of  dimensions  as  given  in  the  figure 
from  A  to  B,  on  which  a  molding,  gutter, 
or  cornice  is  to  be  placed.  Find  the  radius 
of  the  circle  of  which  the  arc  A  NB  is  a  part. 

799.  A,  B  are  two  beacons  on  a  coast- 
line; S  is  a  shoal  off  the  shore,  and  the 
angle  ASB  is  known  to  be  120°.    Show 

that  a  vessel  V  sailing  along  the  coast  will  keep  outside  the  shoal 
if  the  angle  AVB  is  always  less  than  110°. 


208  PLANE  GEOMETRY 

Prove  a  property  of  the  circle  that  you  used  in  proving  that  the 

ship  will  clear  the  shoal. 

.    800.  The  method  given  by  Galileo  for  finding  the  strongest 

rectangular  beam  that  can  be  cut  from  a  round  log  is  as  follows: 
Let  the  circle  A  BCD  represent  the  end  of  the 
log,  and  let  AC  be  a,  diameter.  Divide  the 
diameter  into  three  equal  parts  at  the  points 
M  and  N,  and  from  these  points  erect  perpen- 
diculars intersecting  the  circumference  at 
points  D  and  B.  Draw  AD,  DC,  CB}  and  BA . 
The  rectangle  thus  formed  is  the  cross-section 
of  the  strongest  rectangular  beam.  Show 

that  the  dimensions  of  the  rectangle  are  in  the  ratio  of  1  to  \/2. 
dtSOl.  The  resultant  of  two    forces  is  300  Ibs.     One  of  the 

forces  acting  at  an  angle  of  37°  with  the  resultant  is  100  Ibs. 

Find  the  other  force,  if  the  forces  act  at  an  angle  of  65°  with 

each  other. 

dt802.  The  radius  of  a  circle  is  7  ft.    What  angle  will  a  chord  of 

the  circle  10  ft.  long  subtend  at  the  center? 

803.  To  prolong  a  line  through  an  obstacle  and  to  measure  the 
distance  along  the  line  through  the  obstacle. 

A  and  B  are  two  points  on 
the  given  line.  Take  -X"  any 
point.  Measure  AX.  Take 
XC=AX  and  in  line  with  AX. 
Mark  Y  on  the  mid-point  of 
XC.  Make  DX  =  BX  and  in 
line  with  BX.  MakeZY=YD. 
Make  ZE  =  ZD  and  in  line  with  ZD.  Make  ZF  =  ZC  and  in  line 
with  ZC.  Prove  FE  in  line  with  AB.  What  line  of  the  figure 
equals  BFl 

804.  The  cross-section  of  the  train-shed  of  a  railroad  station  is 
to  have  the  form  of  a  pointed  arch,  made  of  two  circular  arcs,  the 
centers  of  which  are  on  the  ground.    The  radius  of  each  arc  equals 
the  width  of  the  shed,  or  210  ft.    How  long  must  the  supporting 
posts  be  made  which  are  to  reach  from  the  ground  to  the  dome  of 
the  roof? 


THE  CIRCLE 


209 


g805.  Show  by  a  graph  that  the  area  of  a  triangle  having  a 
fixed  altitude  varies  as  the  base,  and  that  one  having  a  fixed 
base  varies  as  the  altitude. 

g806.  Construct  a  graph  showing  the  relation  between  the  areas 
and  the  sides  of  equilateral  triangles. 

g807.  Construct  a  graph  showing  the  relation  between  a  side 
and  the  area  of  a  regular  hexagon.  By  means  of  it  find  the  area  of 
a  regular  hexagon  whose  sides  are  6.  Compare  with  the  computed 
area. 

g808.  If  a  side  of  a  given  regular  polygon  is  a  and  its  perimeter 
p,  graph  the  relation  of  the  perimeters  and  the  corresponding  sides 
of  polygons  similar  to  the  given  polygon. 

g809.  Given  a  polygon  with  area  A  and  a  side  a.  Construct  a 
graph  showing  the  relation  between  the  areas  and  the  sides  corre- 
sponding to  a  in  polygons  similar  to  the  one  given. 

dt810.  The  figure  shows  a 
method  for  determining  the 
horizontal  distance  PR,  and 
the  difference  of  level  QR  be- 
tween two  points  P  and  Q.  A 
rod  with  fixed  marks  A  and 
B  upon  it  is  held  vertical  at 
Q,  and  the  elevation  of  these 
points  ACD  (  =  a)  and  BCD 
(=/3)  are  read  by  a  telescope  and  divided  circle  at  C,  the  axis 
of  the  telescope  being  at  a  distance  CP  ( =  a)  above  the  ground 
at  P.  If  .QA  =  b  and  AB  =  s,  write  down  expressions  for  PR 
(=x)  and  QR  (=y). 

Find  x  and  y  when  a  =  6°  10'  and  0  =  7°  36',  the  values 
of  a,  6,  and  s  being  5  ft.,  2^-  ft.,  and  5  ft. 
respectively., 

dtSll.  Find  the  radius  of  a  parallel  of  lati- 
tude passing  through  Portland,  Me.  (43°  40' 
N.  lat.),  if  the  radius  of  the  earth  is  taken  as 
4000  mi. 

(Note  that  in  the  figure  <&x  equals  <£t/. 
Why?) 

14 


210 


PLANE  GEOMETRY 


812.  The  oval  in  the  figure  is  a  design  used  in  the  construction 
of  sewers.  It  is  constructed  as  follows:  In  the  circle  0  let  CD, 

the  perpendicular  bisector  of  AB,  meet 
AB  at  Oi.  Arcs  AM  and  BN  are  drawn 
with  AB  as  radius  and  with  centers 
B  and  A  respectively. 

The  chords  B0\  and  AOi  meet  these 
arcs  in  M  and  N  respectively. 

The  arc  M  DN  has  the  center  Oi  and 
radius  0\M. 

(a)  Make  an  accurate  construction 
of  the  design. 

(6)  Is  arc  ACB  tangent  to  arc  AM 
and  arc  BN  at  A  and  B  respectively? 
Why? 

(c)  Is  arc  MDN  tangent  to  arc  AM  and  arc  BN  at  M  and  N 
respectively?    Why? 

(d)  If  AB  equals  8  ft.,  find  BOi,  and  hence  OiM,  and  finally  CD. 
That  is,  if  the  sewer  is  8  ft.  wide,  what  is  its  depth? 

d  (e)  If  the  width  of  the  sewer  is  a  ft.,  show  that  its  depth  is 


d  (/)  If  the  depth  of  the  sewer  is  d  ft.,  show  that  its  width  is 


d  (g)  Compute  to  two  places  of  decimals  the  width  of  a  sewer 
whose  depth  is  12  ft. 

813.  The  circumference  of  the  earth  is  approximately  25000 
miles.    Suppose  an  iron  band  25000  miles  long  fits  tightly  around 
it.    If  you  cut  the  band  and  put  in  three  feet,  will  the  band  then  be 
raised  any  appreciable  distance  from  the  earth? 

As  much  as  -J-  inch,  for  example?  T^  inch? 
Consider  the  band,  when  enlarged,  to  be  raised 
an  equal  distance  from  the  earth  at  all  points. 

814.  In  this  design,  the  side  of  the  square 
upon  which  it  is  constructed  is  4a.     Find  the 
area  of  the  shaded  portion. 


THE  CIRCLE 


211 


dt815.  In  constructing  a  sail,  the  amount  of  surface  of  canvas 

ABCD  is  known,  and  the  lengths  of  AB,  AD,  and  DC  are  given. 

The  angle  A  is  a  right  angle.    Show  how  to 

construct  the  angle  between  DC  and  DA. 
dt816.   Find  the  number  of  square  yards 

of  cloth  in  a  conical  tent  with  a  circular 

base  and  the  vertex  angle  72°,  the  center 

pole  being  12  ft.  high. 

817.  A  girder  to  carry  a  bridge  is  in  the 

form  of  a  circular  arc.    The  length  of  the 

span  is  120  ft.,  and  the  height  of  the  arc  is  25 

ft.     Find  the  radius  of  the  circle. 

dt818.  In  the  side  of  a  hill  which  slopes 
upward  at  an  angle  of  32°,  a  tunnel  is  bored 
sloping  downwards  at  an  angle  of  12°  15'  with 
the  horizontal.  How  far  below  the  surface  of 
the  hill  is  a  point  115  ft.  down  the  tunnel? 

819.  In  constructing  a  gas  engine  the  piston 
D,  which  is  in  the  form  of  an  inverted  cup,  is 
5  in.  in  inside  diameter;  the  crank  AB  is  5  in. 
between  the  centers  of  the  pivots,  and  the  con- 
necting rod  AC  is  17  in.  between  the  centers 
of  the  pivots.  How  far  from  the  mouth  of 
the  cup  must  the  pin  C  be  adjusted  in  order 
that  the  connecting  rod  may  just  clear  the 

edge  of  the  cup  at  E  and  F,  the  diameter  of  AC  being  1  in.? 


PART  II 

SECOND  STUDY 


CONTENTS 
PART  TWO— SECOND  STUDY          PAOE 

SUGGESTIONS  FOR  A  REVIEW  OF  THE  FIRST  STUDY 219 

PLATES:     FAMILY  TREES 220,  221,  222,  223,  224 

CHAPTER  I 
FUNDAMENTALS.    RECTILINEAR  FIGURES 

Exercises.    Set  LXXV.    Triangles    229 

Exercises.    Set  LXXVI.    Perpendiculars,  Parallels,  Sums  of  Angles 

of  Polygons 233 

Exercises.    Set  LXXVII.    Parallelograms 235 

Exercises.    Set  LXXVII1.    Inequalities 237 

SUMMARY  OF  TERMS  INTRODUCED 238 

CHAPTER  II 
AREAS  OF  RECTILINEAR  FIGURES 

Exercises.    Set  LXXIX.    Areas 241 

SUMMARY  OF  TERMS  INTRODUCED 242 

CHAPTER  III 
SIMILARITY 

DIVISION  OF  A  SECT 244 

Exercises.    Set  LXXX.    Ratio,  Proportion,  Parallels 245 

Exercises.    Set  LXXXI.    Similarity  of  Triangles 250 

Exercises.    Set  LXXXII.    Similarity  of  Polygons 256 

Exercises.    Set  LXXXIII.    Metric  Relations 262 

CHAPTER  IV 
Locus 

CONCURRENCE 267 

Exercises.    Set  LXXXIV.    Locus 269 

CHAPTER  V 
THE  CIRCLE 

Exercises.    Set  LXXXV.    The  Straight  Line  and  the  Circle 274 

Exercises.    Set  LXXXVI.    Measurement  of  Angles 279 

Exercises.    Set  LXXXVII.    Metric  Relations 286 

Exercises.    Set  LXXXVIII.    Mensuration  of  the  Circle 291 

215 


216  CONTENTS 

CHAPTER  VI 
METHODS  OP  PROOF  PAGE 

A.  DIRECT 297 

Exercises.    Set  LXXXIX.    Synthetic  Methods  of  Proof: 

I.  Geometric 297 

II.  Algebraic 298 

B.  INDIRECT 299 

Exercises.    Set  XC.    Proof  by  the  Method  of  Exclusion 300 

Exercises.    Set  XCI.    Proof  by  Reduction  to  an  Absurdity 301 

Exercises.    Set  XCII.    Analytic  Method  of  Proof 304 

SUGGESTIONS  AS  TO  METHOD  OF  PROCEDURE 305 

CHAPTER  VII 

CONSTRUCTIONS.    METHODS  OP  ATTACKING  PROBLEMS 

Exercises.  Set  XCIII.  Synthetic  Solutions 309 

DISCUSSION  OF  A  PROBLEM 311 

Exercises.  Set  XCIV.  Intersection  of  Loci 312 

FORMAL  ANALYSIS  OF  A  PROBLEM 318 

Exercises.    Set  XCV.    Problems  Calling  for  Analysis : . .  321 

CHAPTER  VIII 
SUMMARIES  AND  APPLICATIONS 

A.  SYLLABUS  OF  THEOREMS 324 

B.  SYLLABUS  OF  CONSTRUCTIONS 334 

C.  SUMMARY  OF  FORMULAS 336 

Z>.  SUMMARY  OF  METHODS  OF  PROOF 338 

Exercises.    Set  XCVI.    Congruence  of  Triangles 339 

Exercises.    Set  XCVII.    Equality  of  Sects 340 

Exercises.    Set  XCVIII.    Equality  of  Angles 340 

Exercises.    Set  XCIX.    Parallelism  of  Lines 341 

Exercises.    Set  C .    Perpendicularity  of  Lines 342 

Exercises.    Set  CI.    Inequality  of  Sects 343 

Exercises.    Set  C1I.    Inequality  of  Angles ' 344 

Exercises.    Set  CHI.    Similarity  of  Triangles 345 

Exercises.    Set  CIV.    Proportionality  of  Sects 346 

Exercises.    Set  CV.    Equality  of  Products  of  Sects .  .  . 346 

Exercises.    Set  CV1.   Miscellaneous  Exercises . .             347 


CONTENTS  217 

CHAPTER  IX 
COLLEGE  ENTRANCE  EXAMINATIONS  PAGE 

CHICAGO 362 

HARVARD 363 

CHAPTER  X 

SUGGESTIONS 

A.  LIST  OP  TOPICS  SUITABLE  FOR  STUDENTS'  DISCUSSION: 

General 374 

Arithmetic 375 

Algebraic 375 

Geometric 375 

B.  TOPICS  WITH  DEFINITE  REFERENCES: 

Geometric  Fallacies 376 

Number  Curiosities 376 

Pythagorean  Proposition 376 

C.  LIST  OF  BOOKS  SUITABLE  FOR  STUDENTS'  READING: 

History  376 

Recreations    377 

Practical    377 

General    378 

Index  of  Definitions .  379 


SUGGESTIONS  FOR  A  REVIEW  OF  THE  FIRST 
STUDY  OF  PLANE  GEOMETRY 

Before  beginning  our  Second  Study  of  Plane  Geometry,  it  might  be  well 
for  us  to  review  the  First  Study.  The  following  material  furnishes  a  brief, 
suggestive  outline  for  such  a  review. 

A.  CONGRUENCE — THEOREMS  1-8 
I.  Define:  * 

Sect,  polygon,  axiom,  postulate,  corollary,  adjacent  angles,  congruent, 

homologous,  perpendicular. 
II.  Summarize: 

a.  Conditions  under  which  triangles  are  congruent  in  general; 

b.  Special  conditions  under  which  right  triangles  are  congruent; 

c.  Facts  about  perpendiculars. 
III.  Family  Trees  of  Propositions: 

To  trace  a  proposition  back  to  its  sources,  that  is,  back  to  the 
definitions,  postulates,  and  axioms  upon  which  it  rests,  will  be  found 
an  interesting  and  profitable  form  of  review.  A  convenient  arrange- 
ment is  to  make  a  "family  tree"  of  a  theorem,  the  branches  of  which 
are  the  authorities  quoted  in  proving  the  proposition.  Each  branch 
should  be  followed  down  as  in  the  main  proposition  until  it  ends  in 
a  postulate,  an  axiom,  or  a  definition. 

Such  a  tree  of  Theorem  6  is  given  as  an  illustration  (Plate  1). 
The  student  is  advised  to  make  a  tree  of  Theorem  5. 

B.  PARALLELS — THEOREMS  9-12 
I.  Define: 

Parallels,  transversal,  alternate-ulterior  angles. 
II.  Classify  angles  according  to: 
a.  Individual  size; 
6.  Relative  size; 
c.  Relative  position. 

III.  Summarize: 

a.  Conditions  under  which  lines  are  parallel; 

b.  Methods  of  proving  sects  equal; 

c.  Methods  of  proving  angles  equal. 

IV.  Family  Tree: 

A  family  tree  of  Theorem  11,  Cor.  2,  is  appended  (Plate  2). 
The  student  is  advised  to  study  it  and  make  one  of  Theorem  12. 

*  Consult  the  First  Study  only  when  necessary. 

210 


220 


PLANE  GEOMETRY 


it 

J 


Is 

I  cS 
J2  *-» 

•a& 


a 


1! 

tr*     a 


es  are 
are  eq 


II 


n 

^  ft 


'&§ 


lologous  parts  of  c 
at  n-gons  are  equ; 


ular  can 
m  a  poiu 


j* 

£ 

a 
O 


£        r9 


0-S  * 


M|l 

H'53 
• 
^•3 

II 


i! 

•3S 


II 

11 

a;  rt 

«5 


— **  o1 
to  ® 

5S 


•rl 


- 


o  a 


REVIEW  OF  FIRST  STUDY 


221 


ill   II 


!°    ^ 

I? 

2   so* 


s  •« 
I  I 


'5 

,  -2 


ha" 
II 


$11     — 


LID 


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v;  g 


-01555 


222  PLANE  GEOMETRY 

C.  SUMS  OF  ANGLES — THEOREMS  13-15 
I.  Define: 

Exterior  angle. 
II.  Summarize: 

The  facts  about  the  sums  of  the  angles,  interior  and  exterior,  of 

an  n-gon. 
III.  Make  a  family  tree  of  Proposition  14. 

D.  PARALLELOGRAMS — THEOREMS  16-21 
I.  Define: 

Parallelogram,  diagonal. 

II.  Classify  quadrilaterals. 

Is  it  possible  to  classify  them  in  more  than  one  way?     If  so,  is  there 
any  preference? 

III.  Summarize: 

a.  Conditions  under  which  a  quadrilateral  is  a  parallelogram; 

b.  Properties  of  a  parallelogram. 

IV.  Make  a  family  tree  of  Proposition  17. 

E.  AREAS — THEOREMS  22-27 
I.  Define: 

Commensurable,  measurement,  area. 
II.  Summarize  the  facts  about  areas  by  giving  formulas  for  the  areas  of 

figures  studied. 
III.  Complete  the  family  tree  of  Theorem  27  as  appended  (Plate  3). 

F.  SIMILARITY— THEOREMS  28-39 
I.  Define. 

Ratio,  proportion,  center  of  similitude,  similar  figures. 

II.  Summarize: 

a.  Properties  arising  from  similarity  of  triangles; 
6.  Conditions  under  which  triangles  are  similar. 

III.  A  family  tree  of  Theorem  39,  Cor.  2,  is  begun  (Plate  4).    The  student  is 
advised  to  make  a  tree  of  Proposition  36. 

G.  Locus — THEOREMS  40-41 
I.  Define: 

Locus  of  a  point. 
II.  State  the  facts  of  which  proofs  are  necessary  and  sufficient  to  establish 

a  locus  theorem. 
III.  State  the  two  locus  theorems  proved  in  the  syllabus. 


•S 

I 


•a  a 


REVIEW  OF  FIRST  STUDY 
PLATE  4 

C°mps'  °f  equal  *  are  equah 


223 


The  altitude  upon  the  hypotenuse  I 

of  a  right  triangle  divides  it  into 

.    .,  ,      ., 

triangles  similar  to  each  other 

,,,,.., 
and  to  the  original. 


n 

Triangles  are  similar  if  two  angles 

e  ,\ 

of  one  are  equal  to  those  of] 

another. 


The  homologous  sides  of  similar 
triangles  have  a  constant  ratio. 


[  The  homologous  angles  of  similar  | 
triangles  are  equal.  \ 

Post,  of  superposition. 

If  corresponding  angles  are  equal,! 
lines  cut  by  a  transversal  are] 
parallel. 

A  line  parallel  to  the  base  of  a 
triangle  cuts  the  remaining  sides 
so  that  they  are  proportional  to 
either  pair  of  homologous  sects.! 

Quantities  equal  to  the  same 
quantity  are  equal  to  each  other. 


The  products  of  equals  multiplied 
by  equals  are  equal. 

The   sums   of   equals   added    to 

equals  are  equal. 
The  whole  is  equal  to  the  sum  of 

its  parts.  (To  be  completed  by  the  student.) 

H.  THE  CIRCLE  AND  STRAIGHT  LINE — THEOREMS  42-49 
I.  Define: 

Circle,  chord,  secant,  tangent. 

II.  Summarize  the  facts  proved  in  the  syllabus  about: 

a.  Chords; 
6.  Tangents; 
c.  Arcs. 

III.  Make  a  family  tree  of  Proposition  48. 

I.  THE  CIRCLE  AND  ANGLE  MEASUREMENT — THEOREMS  50-55 

I.  Define: 

Central  angle,  inscribed  angle. 


224 


PLANE  GEOMETRY 


II.  State  the  method  of  measuring: 
a.  Central  angles; 
6.  Inscribed  angles; 

c.  Angles  formed  by  a  tangent  and  a  chord; 

d.  Angles  with  vertices  outside  the  circle; 

e.  Angles  with  vertices  inside  the  circle. 

III.  A  family  tree  of  Proposition  55  is  begun  (Plate  5).    The  student  should 
make  one  of  Theorem  53. 

J.  MENSURATION  OF  THE  CIRCLE — THEOREMS  56-64 

I.  Define: 

Sector,  apothem. 
II.  State  formulas  for: 

a.  Area  of  a  regular  polygon; 

b.  Circumference  of  a  circle; 

c.  Area  of  a  circle. 

III.  Summarize  the  methods  now  known  to  you  of  proving: 
a.  Sects  equal; 
6.  Angles  equal; 

c.  Lines  perpendicular; 

d.  Lines  parallel; 

e.  Triangles  congruent; 
/.  Triangles  similar; 

g.  Arcs  equal; 
i.   Chords  equal. 

PLATE  5 

Th.  55.    A  tangent  from  a  point  to  a  circle  is  the  mean 
proportional  between  any  secant  and  its  external  sect,  from 
the  same  point  to  the  circle. 


1 

A.    An    inscribed 

1 

Any  two  quanti- 

I               l 

B.    Triangles   are     C.     Homologous 

angle,      or      one 

ties  of  the  same 

similar  when  two    sides    of    similar 

formed  by  a  tan- 

kind compare  as 

angles  of  one  are     triangles  have   a 

gent  and  a  chord  is 

their   numeric 

equal  each  to  each     constant  ratio. 

measured  by  one- 

measures. 

to  two  angles  of                  | 

half  its  intercepted 

another. 

arc. 

1 

NOTE. — If  the  student  is  in  need  of  further  review,  he  is  advised  to  trace 
branches  A,  B,  and  C  of  this  family  tree  back  to  their  sources. 


CHAPTER  I 


FUNDAMENTALS.    RECTILINEAR  FIGURES 

Theorem  1.    Vertical  angles  are  equal. 

The  pupil  is  earnestly  advised  not  to  refer  to  the  First  Study* 
for  suggestions  as  to  proofs  of  the  theorems  there  taken  up  if  it 
is  possible  for  him  to  work  them  out  without  any  help  in  the  review. 
For  this  reason  those  theorems  are  stated  in  this  part. 

Theorem  2.  Two  sides  and  the  included  angle  determine  a 
triangle. 

Given:    AABC  and  ADEF]  AB  =DE]  %.B  =  %.E;  and  BC  =EF. 
Prove:    AABC&ADEF. 

E 


(1)  Place  ADEF  on  AABC  so  that 
%.E  coincides  with  ^.B  and  ED  falls 
along  BA. 

(2)  Then    EF    falls     along     BC. 
'.'  4#=3_£. 

V  ED  =  BA,  D  will  fall  on  A. 
'.'  EF  =  BC,  F  will  fall  on  C. 

(3)  .'.  FD  coincides  with  CA. 

(4)  .'.  ADEF  ¥  AABC. 


PROOF 

(1)  Superposition  post. 


(2)  Data. 


(3)  Two  points  fix  a  straight  line. 

(4)  Def.  of  congruence. 


*  Throughout  "A  Second  Study  of  Plane  Geometry"  reference  will  be 
made  to  "A  First  Study  of  Plane  Geometry"  as  "First  Study,"  and  no  proofs 
will  be  given  in  this  part  of  propositions  contained  in  the  former  except  in  the 
case  of  the  congruence  of  triangles.  In  this  instance,  one  proof  by  the  method 
of  superposition  will  be  given  in  full,  since  cutting  and  pasting  were  used 
in  the  First  Study  owing  to  the  difficulty  of  the  usual  proof  for  the  beginner. 
15  225 


226 


PLANE  GEOMETRY 


Theorem  3.    Two  angles  and  the  included  side  determine  a 
triangle. 

Apply  the  method  of  proof  used  in  Theorem  2. 

What  parts  of  the  triangles  will  you  first  make  coincident  in  superposing 
in  this  case? 

What  postulate  is  needed  in  order  to  clinch  this  proof? 

Theorem  4.    The  bisector  of  the  vertex  angle  of  an  isosceles 
triangle  divides  it  into  two  congruent  triangles. 

Cor.  1.  The  angles  opposite  the  equal  sides  of  an  isosceles 
triangle  are  equal. 

Cor.  2.  The  bisector  of  the  vertex  angle  of  an  isosceles  tri- 
angle bisects  the  base  and  is  perpendicular  to  it. 

Cor.  3.    An  equilateral  triangle  is  equiangular. 

Cor.  4.  The  bisectors  of  the  angles  of  an  equilateral  triangle 
bisect  the  opposite  sides  and  are  perpendicular  to  them. 

Cor.  5.  The  bisectors  of  the  angles  of  an  equilateral  triangle 
are  equal. 

Theorem  5.    A  triangle  is  determined  by  its  sides. 

Theorem  5a.    Only  one  perpendicular  can  be 

drawn  through  a  given  point  to  a  given  line. 

Given:    (I)  Point  P  in  'AB;  (II)  Point  P  outside  AB. 

Prove:  Only  one  line  through  Pis  perpendicular 
to  AB. 

Suggestion:  (I)  When  P  is  in  AB,  in  how  many 
positions  of  PD  will  the  st.  ^.APB  be  di- 
vided into  two  right  2$_s? 

PROOF  (II) 

(1)  Suppose  PD  meets  ~AB  at  rt.  4s 
at  D,  and  PC  is  a  line  drawn  to  any 
other  point  C  in  AB. 

(2)  Extend  PD  to  PI  so  that  PiD 


D 


\ 


(3)  PCPi  is  not  a  straight  line. 

(4)  /.  4PCPiisnotast.2$_. 

(5)  APCD^APiCD. 

(6)  :.  ^PCD^^PiCD. 

(7)  .'.  4PCD  is  not  a  rt.  %.. 

(8)  .'.  PC±AB. 


(3)  Why? 

(4)  Why? 

(5)  Why? 

(6)  Why? 

(7)  Why? 

(8)  Why? 


FUNDAMENTALS.    RECTILINEAR  FIGURES      227 

Theorem  5fr.  Two  sects  drawn  from  a  point  in  a  perpendicular 
to  a  given  line,  cutting  off  on  the  given  line  equal  sects  from 
the  foot  of  the  perpendicular,  are 
equal  and  make  equal  angles  with 
the  perpendicular. 

Given:   PQA.AB  at  Q;  P  any  point  in 

PQ',  QC=QD;  C  and  D  in  AB. 

Prove:   PC=PD;  ^CPQ^^DPQ.  __^^_*^_A__*___Jk_ 

(Proof  left  to  the  student.)  AC  Q  D      B 

In  our  first  study,  propositions  dealing  with  inequalities  were 
with  one  exception  omitted.  The  following  group  of  propositions 
supplies  this  omission,  and  before  proving  them  it  will  be  necessary 
to  study  a  set  of  axioms  dealing  with  inequalities. 

1.  If  unequals  are  operated  on  in  the  same  way  by  positive  equals, 
the  results  are  unequal  in  the  same  order. 

e.g.,  If  a>b  and  c=d,  where  c  and  d  are  positive  quantities,  a+c> 

b+d,  a-c>b  -d,  ac>bd}  ->  -;  while  ae>bd  and  Va> %/b  under 

c    a 

certain  conditions  which  do  not  affect  work  in  elementary  geometry. 

Test  these  statements,  substituting  for  the  symbols  the  following 
values : 

(1)  a  =  5,  6  =  3,  c  =  d  =  4.  (2)  a=£,  &=£,  c  =  d=4.       - 

(3)  a  =  Z,  6=  -Z,  c  =  d  =  4.       (4)  a=|-,  6=  --J-,  c  =  d  =  3. 
(5)  a=  -sV,6=  -i,c  =  d  =  3. 

What  conclusions  can  you  draw? 

What  meaning  is  attached  to  the  phrase  "in  the  same  order"? 

2.  The  sums  of  unequals  added  to  unequals  in  the  same  order  (or 
the  same  sense)  are  unequal  in  the  same  order. 

e.g.,  If  a>b,  and  c>d,  then  a+c>b+d. 

Show  why  this  statement  could  not  be  made  for  subtraction  of 
unequals. 

3.  The  differences  of  unequals  subtracted  from  equals  are  unequal 
in  the  reverse  order. 

Illustrate  this  fact. 

4.  //  the  first  of  three  quantities  is  greater  than  the  second,  which 
in  turn  is  greater  than  the  third,  all  the  more  then  is  the  first  greater 
than  the  third. 

Illustrate  this  fact. 


228 


PLANE  GEOMETRY 


Theorem  5c.    The  sum  of  two  sects  drawn  from  any  point 

inside  a  triangle  to  the  ends  of  one  of 
its  sides  is  less  than  the  sum  of  its 
remaining  sides. 

Given:    AABC;  D  any  point  inside. 
Prove:   DA+DC<BA+BC. 

PROOF 
Extend  AD  to  E  in  EC. 

(1)  DA+DE<what  part  of  AB+        (1)  Why? 
BCt  DC<DE+ whatpartof A  B + £C? 

(2)  /.  DA+Z>#+DC<what?  (2)  Why? 

(3)  /.  DA+DC<BA+BC.  (3)  Why? 

Theorem  5d.  Two  sects  drawn  from  a  point  in  a  perpendicular 
to  a  given  line  and  cutting  off  p 

unequal  distances  from  the  foot  of 
the  perpendicular  are  unequal  in 
the  same  order  as  those  distances, 
and  conversely. 

I.  Direct.    

Given:   PC LAQCRB;  CR>CQ. 
Prove:   PR>PQ. 

Suggestions :   Where  will  Qi  lie  with  res- 
pect to  C  and  R  if  CQi  =  CQt  Why? 

Why  is  it  that  whatever  you  prove  true  of  PQi  is  true  of  PQ? 
Why  is  it  that  whatever  is  true  of  PQi+PiQi  and  PR+PtR  is  true  of 

PQi  and  Pfl? 
Write  a  complete  proof  of  this  theorem. 

II.  Converse. 

Given:   PC JLAQCRB;  PR>PQ. 

Prove:   CR>CQ. 

Suggestions:   Why  can  CR  not  be  less  than  CQ? 

Why  can  CR  not  equal  CQ? 

What  remains  for  the  relation  of  CR  to  CQ? 

For  convenience  PQi+QiPi  is  at  times  referred  to  as  PQiPi  and 
is  called  a  broken  line.  Such  a  line  is  always  composed  of  two  or 
more  sects. 

The  method  of  proof  here  outlined  is  known  as  the  method  of 
exclusion  or  elimination.  Any  two  quantities  of  the  same  kind 
(a  and  b)  must  bear  one  of  the  following  relations  to  each  other: 


FUNDAMENTALS.    RECTILINEAR  FIGURES      229 

a  >  6,  a  =  b,  or  a  <  b.  If  any  two  of  these  relations  can  be  shown  to 
be  false,  the  remaining  relation  must  therefore  be  true.  For 
further  discussion  and  illustration  see  p.  299. 

Cor.  1.  All  possible  obliques  from  a  point  to  a  line  are  equal 
in  pairs,  and  each  pair  cuts  off  equal  sects  from  the  foot 
of  the  perpendicular  from  that  point  to  the  line. 

(I)  Under  what  condition  will  obliques  from  a  point  to  a  line  be  equal  ? 

Why,  then  will  all  possible  obliques  from  a  point  to  a  line  be  equal  in 
pairs? 

(II)  Use    the  method  of   exclusion  to    prove  the    second    part  of   the 
corollary. 

Theorem  6.    The  perpendicular  is  the  shortest  sect  from  a 
point  to  a  line. 

Suggestion:  Use  Theorem  5d  to  give  a  much  simpler  proof  than  was 
possible  in  the  First  Study. 

Theorem  6a.    The  shortest  sect  from  a  point  to  a  line  is  perpen- 
dicular to  it. 

Hint:  Show  that  this  is  a  special  case  of  Theorem  5d  (converse) 

Theorem  7.    The  hypotenuse  and  adjacent  angle  determine  a 
right  triangle. 

Theorem  8.    The  hypotenuse  and  another  side  determine  a 
right  triangle. 

EXERCISES.     SET  LXXV.    TRIANGLES 
Numeric 

820.  The  perimeter  of  an  isosceles  triangle  is  13,  and  the  ratio 
of  one  of  the  equal  sides  to  the  base  is  1%.    Find  the  three  sides. 

Theoretic 

821.  In  proving  triangles  congruent,  two  methods  have  been 
used. 

(a)  When  what  elements  are  given  equal  can  superposition  be 
used? 

(b)  When  must  juxtaposition  be  used?    Answer  in  a  single  brief 
sentence. 

822.  The  sects  of  any  bisector  of  a  given  sect  cut  off  by  per- 
pendiculars erected  at  the  ends  of  the  given  sect  are  equal. 


230 


PLANE  GEOMETRY 


823.  The  sects  of  a  perpendicular  to  the  bisector  of  an  angle  at 
any  point  in  it  and  limited  by  the  sides  of  the  angle  are  equal. 

824.  Sects  joining  any  point  in  the  bisector  of  an  angle  to  points 
on  the  sides  equidistant  from  the  vertex  are  equal. 

825.  If  in    the    accompanying   diagram, 
$.BAC  =  $BCA, and  $BAY=  $BCX,  sect 
ZC  =  sect  YA. 

826.  If  the  sides  of  an  equilateral  triangle 
are  prolonged  in  turn  by  equal  lengths,  and 
the  extremities  of  these  sects   are  joined, 
another  equilateral  triangle  is  formed. 

827.  Two  isosceles  triangles  are  congru- 
ent if  one  of  the    equal    sides    and    the 
altitude  upon  that  side  are  equal  each  to  each. 

828.  Triangles  are  congruent  if  two  sides  and  the  altitudes  upon 
one  of  them  are  equal  each  to  each. 

829.  A  triangle  is  determined  by  a  side  and  the  median*  and 
the  altitude  to  that  side.  (X 

d830.  If  in  the  accompanying  diagram 
%A  is  a  right  angle,  YX=AC,  CP  =  PY, 
show  that  PB+PX>CB+CA. 

831.  The  sum  of  the  distances  from 

any  point  inside  a  triangle  to  the  vertices    A  X  B 

is  greater  than  its  semiperimeter,  but  less  than  its  perimeter. 

832.  The  sum  of  the  diagonals  of  a  quadrilateral  is  greater  than 
the  sum  of  either  pair  of  opposite  sides. 

833.  The  sum  of  the  diagonals  of  a  quadrilateral  is  less  than  its 
perimeter,  but  greater  than  its  semiperimeter. 

834.  The  sum  of  the  medians  of  a  triangle  is  less  than  one  and  a 
half  times  its  perimeter.    (Prove  this  and  the  following  exercise 
without  assuming  the  concurrence  of  the  medians). 

835.  The  sum  of  the  medians  of  a  triangle  is  greater  than  its 
semiperimeter. 

836.  Each  altitude  of  a  triangle  is  less  than  half  the  sum  of  the 
adjacent  sides. 

*A  median  is  the  sect  between  any  vertex  of  a  A  and  the  midpoint  of 
the  side  opposite  that  vertex. 


FUNDAMENTALS.     RECTILINEAR  FIGURES      231 

837.  The  sum  of  the  altitudes  of  a  triangle  is  less  than  the 
perimeter. 

838.  Show  that  the  bisector  of  the  vertex  angle  of  an  isosceles 
triangle  is  coincident  with  the  altitude  to  its  base. 

Construction  * 

839.  Bisect  a  reflex  angle. 

d840.  To  construct  a  triangle,  having  given  a  side,  the  median 
and  the  altitude  to  a  second  side. 

d841.  To  construct  a  triangle  having  given  one  side,  the  corre- 
sponding median,  and  the  altitude  to  another  side. 

Theorem  9.    Lines  perpendicular  to  the  same  line  are  parallel. 

Theorem  10.  A  line  perpendicular  to  one  of  a  series  of  parallels 
is  perpendicular  to  the  others. 

Theorem  11.  //  when  lines  are  cut  by  a  transversal  the  alter- 
nate-interior angles  are  equal,  the  lines  thus  cut  are  parallel. 

Cor.  1.  //  the  alternate-exterior  angles  or  corresponding  angles 
are  equal  when  lines  are  cut  by  a  transversal,  the  lines 
thus  cut  are  parallel. 

Cor.  2.  //  either  the  consecutive-interior  angles  or  the  consecu- 
tive-exterior angles  are  supplementary  when  lines  are 
cut  by  a  transversal,  the  lines  thus  cut  are  parallel. 

Theorem  12.  Parallels  cut  by  a  transversal  form  equal  alter- 
nate-interior angles. 

Cor.  1.  Parallels  cut  by  a  transversal  form  equal  corresponding 
angles  and  equal  alternate-exterior  angles. 

Cor.  2.  Parallels  cut  by  a  transversal  form  supplementary 
consecutive-interior  angles  and  supplementary  con- 
secutive-exterior angles. 


*  For  a  discussion  of  methods  of  attacking  problems  in  construction  see 
Chapter  VII,  p.  306,  where  additional  exercises  will  also  be  found.  At  this 
point,  for  instance,  the  topics,  "The  Synthetic  Method  of  Attacking  a  Problem" 
(p.  309)  and  "The  Formal  Analysis  of  a  Problem"  (p.  318)  should  be  studied. 


232  PLANE  GEOMETRY 

Theorem  12a.  Two  angles  whose  sides  are  parallel  each  to  each 
or  perpendicular  each  to  each  are  either  equal  or  supplementary. 

,C  Given:   (I)  X^YX  \\  BA',  Z^YZ 

/<*         \\BC. 
(ID 


A. _^__  ±BC 

$  Z/  Prove:     (I) 

X,  fr  X 


L>        /  ™ 

AI 

180°- 

Suggestions  :   Using  the  con- 
struction lines  prove  (I). 
(II)  If  YiCi  ||  EC  and  YiAi\\BA,  what  relation  exists  between  %.ABC 

and  2$f.AiYiCit 

How  are  YiCi  and  YiZi  related?    How  YiAi  and  Y\Xi! 
Then  how  are  ^  AiYiCi  and  ZiYiAi  related? 
Then  how  are  %?  XiYiZi  and  Z\Y\A\  related? 
Draw  conclusions. 

Theorem  13.  The  sum  of  the  angles  of  a  triangle  is  a  straight 
angle. 

Cor.  1.    A  triangle  can  have  but  one  right  or  one  obtuse  angle. 

Cor.  2.  Triangles  having  two  angles  mutually  equal  are  mutu- 
ally equiangular. 

Cor.  3.  A  triangle  is  determined  by  a  side  and  any  two  homolo- 
gous angles. 

Cor.  4.  An  exterior  angle  of  a  triangle  is  equal  to  the  sum  of 
the  non-adjacent  interior  angles. 

Theorem  14.  The  sum  of  the  angles  of  a  polygon  is  equal  to  a 
straight  angle  taken  as  many  times  less  two  as  the  polygon  has 
sides. 

Cor.  1.    Each  angle  of  an  equiangular  polygon  of  n  sides  equals 

the  —^—th  part  of  a  straight  angle, 
n 

Cor.  2.    The  sum  of  the  exterior  angles  of  a  polygon  is  two 

straight  angles. 
Cor.  3.    Each  exterior  angle  of  an  equiangular  polygon  of  n 

sides  is  equal  to  the  -th  part  of  a  straight  angle. 


FUNDAMENTALS.    RECTILINEAR  FIGURES      233 

Theorem  15.    If  two  angles  of  a  triangle  are  equal,  the  sides 
opposite  them  are  equal. 
Cor.  1.    Equiangular  triangles  are  equilateral. 

EXERCISES.      SET    LXXVI.      PERPENDICULARS.      PARALLELS. 
SUMS  OF  ANGLES  OF  POLYGONS 

Numeric 

842.  Find  the  angles  of  an  isosceles  triangle  if  a  base  angle  is 
double  the  vertex  angle. 

843.  If  the  vertex  angle  of  the  isosceles  triangle  is  30°,  find  the 
angle  formed  by  the  bisectors  of  the  base  angles.    If  the  vertex 
angle  is  J5? 

844.  The  bisector  of  the  base  angle  of  an  isosceles  triangle  makes 
with  the  opposite  leg  an  angle  of  53°  17'.    Find  the  angles  of  the 
triangle. 

845.  Find  the  angles  of  an  isosceles  triangle  if  the  altitude  is 
one-half  the  base. 

846.  If  the  angle  at  the  vertex  of  an  isosceles  triangle  is  36°, 
the  bisector  of  a  base  angle  divides  the  triangle  into  two  isosceles 
triangles.    Find  the  lengths  of  all  the  sects  in  the  diagram  if  a  leg 
of  the  given  triangle  is  a  and  the  base  is  b. 

847.  What  is  the  sum  of  the  angles  of  (a)  a  hexagon,  (6)  a  hepta- 
gon, (c)  an  octagon,  (d)  a  nonagon,  (e)  a  decagon,  (/)  a  polygon  of 
18  sides,  (g)  a  polygon  of  24  sides,  (h)  of  30  sides? 

848.  Find  each  angle  of  an  equiangular  (a)  hexagon,  (6)  hepta- 
gon, (c)  octagon,  (d)  nonagon,  (e)  decagon. 

849.  In  what  polygon  is  the  sum  of  the  angles  three  times  as 
great  as  in  a  pentagon. 

850.  How  many  sides  has  a  polygon  if 

(a)  the  sum  of  the  interior  angles  equals  4  rt.<&?    3  st.  •£•? 
6  rt.  <fr?    8  st.  <£?    20  rt.  <£? 

(b)  the  sum  of  the  interior  angles  is  2  (or  3,  or  4,  or  5,  or  6) 
times  as  large  as  the  sum  of  the  exterior  angles? 

(c)  the  sum  of  the  interior  angles  exceeds  the  sum  of  the  exterior 
angles  by  4  rt.  <£*?    3  st.  <fr?    9  st.  «£*? 

(d)  the  ratio  of  each  interior  angle  to  its  adjacent  exterior 
angle  is  2  to  1?    3  to  2?    5  to  1?    a  to  6?    May  a  and  b  have  any 
values  whatever? 


234  PLANE  GEOMETRY 

(e)  each  exterior  angle  contains  40°?    30°?    20°?    120°? 
(/)  each  interior  angle  is  %  st.  <£?   f  st.  <£?   art.  <£?   f  rt.  <£? 
frt.  <£?   fst.  ^ 

Only  what  kind  of  polygon  is  considered  in  (d),  (e), 
and  (/)? 

d851.  If  the  sides  of  a  polygon  are  extended  until  they  intersect, 
a  star  polygon  results.    A  five-pointed  star  is  called  a  pentagram, 
and  is  of  historic  interest  as  the  badge  chosen 
by  the  followers  of  Pythagoras.  Star  polygons 
may  also  be  formed  by  chords  of  circles  or 
by  certain  combinations  of  polygons.    What 
is  the  sum  of  the  vertex  angles  of  a  five- 
pointed    star?    A   six-pointed   star?    An  n- 
pointed  star?    (Make  your  solution  general.) 
d852.  If  A,  B,  C,  denote  the  angles  of  a  triangle,  ha,  hb,  the 
altitudes  upon  BC,  and  AC,  ta,  tb,  the  bisectors  of  A  and  B,  find: 

(a)  <£U&.  (&)  *hjib.  (c)  £a«. 

853.  The  number  of  all  diagonals  of  a  polygon  of  n  sides  is 
— Test  this  statement  in  several  instances. 

,  Theoretic 

854.  The  bisectors  of  a  pair  of  consecutive-interior  angles  of 
parallels  cut  by  a  transversal  are  perpendicular  to  each  other. 

855.  The  bisectors  of  a  pair  of  alternate-interior  angles  of  paral- 
lels cut  by  a  transversal  are  parallel  to  each  other. 

856.  The  bisector  of  an  exterior  angle  at  the  vertex  of  an  isosceles 
triangle  is  parallel  to  the  base,  and  conversely. 

857.  An  exterior  angle  at  the  vertex  of  an  isosceles  triangle  is 
double  a  base  angle. 

858.  State  and  prove  the  converse  of  Ex.  857 

d859.  If  a  leg  of  an  isosceles  triangle  is  produced  through  the 
vertex  by  its  own  length,  and  its  extremity  joined  to  the  extremity 
of  the  base,  the  joining  line  is  perpendicular  to  the  base. 

d860.  The  bisectors  of  the  angles  of  a  quadrilateral  form  a 
quadrilateral  the  sum  of  whose  opposite  angles  is  equal  to  two 
right  angles. 


FUNDAMENTALS.    RECTILINEAR  FIGURES      235 

Theorem  16.    Either  diagonal  of  a  parallelogram  bisects  it. 

Cor.  1.  The  parallel  sides  of  a  parallelogram  are  equal,  and 
the  opposite  angles  are  equal. 

Cor.  2.    Parallels  are  everywhere  equidistant. 

Theorem  17.  A  quadrilateral  whose  opposite  sides  are  equal 
is  a  parallelogram. 

Theorem  18.  A  quadrilateral  having  a  pair  of  sides  both  equal 
and  parallel  is  a  parallelogram. 

Theorem  19.  A  parallelogram  is  determined  by  two  adjacent 
sides  and  an  angle;  or  parallelograms  are  congruent  if  two  adjacent 
sides  and  an  angle  are  equal  each  to  each. 

Theorem  20.  The  diagonals  of  a  parallelogram  bisect  each 
other. 

Theorem  21.  A  quadrilateral  whose  diagonals  bisect  each  other 
is  a  parallelogram. 

EXERCISES.    SET  LXXVII.    PARALLELOGRAMS 
Theoretic 

861.  If  in  the  accompanying  diagram,  DE  \\  FG  \\  HK  \ \  BC  and 
KL\\OH  [\EF\\AB. 

(a)  What  is  the  sum  of  AD,  DE,  EF,  FG,  GH,  HK,  KL, 
andLC? 

(6)  How,  if  at  all,  does 
the  length  of  AC  affect  the 
solution? 

(c)  How,  if  at  all,  does  the 

number  of  parallels  affect  the 

solution?  A        E  G          K.         C 

d862.  The  sum  of  the  perpendiculars  from  any  point  in  the  base 
of  an  isosceles  triangle  to  the  legs  is  constant,  and  equal  to  the 
altitude  upon  a  leg. 

863.  The  sum  of  the  perpendiculars  from  any  point  inside  an 
equilateral  triangle  to  the  three  sides  is  equal  to  the  altitude. 

The  following  group  of  six  propositions  further  supplies  facts 
relating  to  inequalities  of  sects  and  angles  omitted  in  the  First 
Study. 


236 


PLANE  GEOMETRY 


Theorem  21a.     The  difference  between  any  two  sides  of  a 
triangle  is  less  than  the  third  side. 

Hint:   If  c  <a+6,  how  can  we  express  a  in  terms  of  6  and  c? 

Theorem  21&.    If  two  sides  of  a  triangle  are  unequal,  the  angles 
opposite  them  are  unequal  in  the  same  order. 

Suggestions:     Make     ACi=AC.     Justify     this     con- 
struction. 

What  relation  exists  between  4!  and  42? 
What  relation  exists  between  42  and  43? 
B*~* ™"\?         What  relation  exists  between  43  and  4C? 

Theorem  21c.    //  two  angles  of  a  triangle  are  unequal,  the  sides 
opposite  them  are  unequal  in  the 
same  order. 

Suggestions:  Make  4.1  =  45.    Why? 

What   relation     exists    between    AB, 

AAi+AiC,  and^C? 
NOTE. — Since  Prop.  21c  is  the  converse 
of  Prop.  216  an  indirect  proof  would  have 
been  possible.  Why?  Can  you  see  any 
advantage  in  giving  a  direct  proof  where 
convenient? 


C 


Theorem  21c?.  If  two  triangles  have  two  sides  equal  each  to  each 

but  the  included  angles  unequal, 
their  third  sides  are  unequal  in  the 
same  order  as  those  angles. 

Suggestions:  Why  is  it  desirable  and  why 
possible  that    the  triangles  should  be 
placed    together   as  suggested  by  the 
accompanying  diagram  with  AB  (<AC) 
in  coincidence  in  the  two  triangles? 
If  we  could  break  the  sect  BC  into  two 
sects  such  as  BP+PCi  it  would  be 
evident  that  BP+PCi>BCi. 
Then  our  problem  is  to  so  locate  P  that  PCi=PC. 
What  kind  of  triangle,  therefore,  is 
How,  then,  shall  we  draw  CiP? 


FUNDAMENTALS.    RECTILINEAR  FIGURES      237 


Theorem  21e.    If  two  triangles  have  two  sides  equal  each  to 
each,  but  the  third  sides  unequal,   the  A 

angles  opposite  those  sides  are  unequal 
in  the  same  order. 
Suggestions:   Use  method  of  exclusion. 

Theorem  21/.  If  one  acute  angle  of  a 
right  triangle  is  double  the  other,  the 
hypotenuse  is  double  the  shorter  leg,  and 
conversely. 

Suggestions:    (a)  Direct. 

What  part  of  a  straight  angle  is  2a? 
What  kind  of  triangle  is  ACCJ 
How  are  the  sects  CB,  CCi,  and  AC  related? 
(6)  Converse. 

How  is  ACi  related  to  AC? 
What  kind  of  triangle  is  ACCi? 
How  are  the  angles  CiAC,  BAG,  and  ACCi 
related? 

EXERCISES.     SET  LXXVIII.     INEQUALITIES 

Numeric 

864.  Two  sides  of  a  triangle  are  10"  and  13".    Between  what 
limits  must  the  third  side  lie? 

865.  If  two  angles  of  a  triangle    are    respectively    55°  and 
65°,  which   is   the   longest,  and  which  the  shortest  side  of  the 
triangle? 

866.  If  one  angle  of  a  triangle  is  one-third  of  a  straight  angle, 
and  a  non-adjacent  exterior  angle  of  the  triangle  is  five-eighths  of  a 
straight  angle,  which  side  of  the  triangle  is  the  longest  and  which 
is  the  shortest? 

Theoretic 

867.  Either  leg  of  an  isosceles  triangle  is  greater  than  a  sect 
connecting  the  vertex  with  any  point  in  the  base. 

868.  The  sect  joining  the  vertex  of  an  isosceles  triangle  to  any 
point  in  the  prolongation  of  its  base  is  greater  than  either  leg. 

869.  If  one   leg  AB   of   an   isosceles   triangle   ABC   is   pro- 
duced beyond  the  base  BC  to  a  point  D,  then  <£ACD  is  greater 
than 


238  PLANE  GEOMETRY 

870.  If  PQ=QR  in  the  accompanying  diagram,  prove  EP>ER. 

871.  If   no    median    of    a  triangle   is 
perpendicular  to  the  side  to  which  it  is 
drawn,  the  triangle  is  not  isosceles. 

872.  If  any  point  in  the  prolongation 
of  a  leg  produced  through  the  vertex  of 
an  isosceles  triangle  whose  base  is  shorter 

r  „          than  a  leg  is  joined  to  the  extremity  of 

the  base,  a  scalene  triangle  will  be  formed. 

873.  If  a  vertex  of  an  equilateral  triangle  is  joined  to  any  point 
in  the  prolongation  of  the  opposite  side,  a  scalene  triangle  is  formed. 
Which  is  the  longest  and  which  the  shortest  side  of  each  of  the 
triangles  thus  formed? 

d874.  Prove  that  the  sect  joining  an  extremity  of  the  base  of  an 
isosceles  triangle  to  any  point  in  the  opposite  leg  is  greater  than 
one  sect  cut  off  on  that  leg.  Is  it  ever  greater  than  either  sect? 
Under  what  condition  is  it  less  than  one  of  the  sects? 

875.  The  diagonals  of  a  rhomboid  intersect  obliquely,  and  the 
greater  angle  formed  by  them  lies  opposite  the  greater  side  of  the 
parallelogram.     (A  rhomboid  is  an  oblique  O  in  which  the  ad- 
jacent sides  are  unequal.) 

876.  If  a  triangle  is  not  isosceles  the  median  to  any  side  is  not 
perpendicular  to  it,  and  the  larger  angle  which  it  forms  with  that 
side  lies  opposite  the  greater  of  the  remaining  two  sides. 

d877.  Any  point  not  on  the  perpendicular  bisector  of  a  sect  is 
unequally  distant  from  the  ends  of  the  sect. 

LIST  OF  WORDS  DEFINED  IN  CHAPTER  I 

Exclusion,  elimination,  broken  line. 

SUMMARY  OF  AXIOMS  IN  CHAPTER  I 

1.  If  unequals  are  operated  on  in  the  same  way  by  positive  equals,  the 
results  are  unequal  in  the  same  order. 

2.  The  sums  of  unequals  added  to  unequals  in  the  same  order  (or  the 
same  sense)  are  unequal  in  the  same  order. 

3.  The  differences  of  unequals  subtracted  from  equals  are  unequal  in  the 
reverse  order. 

4.  If  the  first  of  three  quantities  is  greater  than  the  second,  which  in  turn 
is  greater  than  the  third,  all  the  more  then  is  the  first  greater  than  the  third. 

(For  a  summary  of  theorems  see  Chapter  VIII,  p.  324.) 


CHAPTER  II 

AREAS  OF  RECTILINEAR  FIGURES 

Theorem  22.  Rectangles  having  a  dimension  of  one  equal  to 
that  of  another  compare  as  their  remaining  dimensions. 

Theorem  23.  Any  two  rectangles  compare  as  the  products  of 
their  dimensions. 

Theorem  24.  The  area  of  a  rectangle  is  equal  to  the  product 
of  its  base  and  altitude. 

Theorem  25.  The  area  of  a  parallelogram  is  equal  to  the 
product  of  its  base  and  altitude. 

Cor.  1.  Any  two  parallelograms  compare  as  the  products  of 
their  bases  and  altitudes. 

In  proving  Cors.  1,  2,  3  of  this  theorem  and  the  next,  express 
the  areas  as  algebraic  formulas  and  apply  axioms.  Why  is  such  a 
procedure  both  convenient  and  natural? 

Cor.  2.  Parallelograms  having  one  dimension  equal  compare 
as  the  remaining  dimensions. 

Cor.  3.  Parallelograms  having  equal  bases  and  equal  altitudes 
are  equal. 

Theorem  26.  The  area  of  a  triangle  is  equal  to  half  the  product 
of  its  base  and  its  altitude. 

Cor.  1.  Any  two  triangles  compare  as  the  products  of  their 
bases  and  altitudes. 

Cor.  2.    Triangles   having  one  dimension  equal  compare  as 

their  remaining  dimensions. 
Cor.  3.   Triangles  having  equal  bases  and  equal  altitudes  are 

equal. 

Theorem  26a.  The  square  on  the  hypotenuse  of  a  right  tri- 
angle equals  the  sum  of  the  squares  on  the  two  legs. 

NOTE. — In  this  text  the  conventional  phraseology  will  be  followed,  and  the 
usual  distinction  between  such  expressions  as  "square  on"  and  "square  of" 
will  be  observed.  Square  on  will  mean  the  area  of  the  square  constructed  on  the 
given  sect,  and  square  of,  the  square  of  the  numeric  measure  of  the  given  sect. 

239 


240 


PLANE  GEOMETRY 


Given:  AABC;  2^BCA=Ti. 4;  squares 
CBFG,  ABDE,  and  CAKH. 

Prove:  Sq.  ABDE  =Sq.  CBFG  +  Sq. 
CAKH. 

Proof:  (Some  steps  which  the  student 
can  readily  supply  have  been  pur- 
posely omitted.) 

Draw  CM  .LEWD. 
Draw  CD  and  AF. 

(1)  .'.  CM  ||£Dand  AE. 

(2)  ACG  is  a  st.  line-_ 

(3)  AABF  =  y2  (BF-CB). 

(4)  AABF=y2  (Sq.  CB/^G). 

(5)  LMDB  is  a  rect.  _ 

(6)  ACDB=^  (BD-LB}. 

(7)  ACDBs^  (Rect.  LMDB}. 

(8)  BD=AB,CB=BF. 

(10)  /.  AABF^ACBD. 

(11)  /.Sq.  CF^Rect.  LD. 


(1)  Why? 

(2)  Why? 

(3)  Why? 

(4)  Why? 

(5)  Why? 

(6)  Why? 

(7)  Why? 

(8)  Why? 

(9)  Why? 

(10)  Why? 

(11)  Why? 


\ 

/" 

\  x 

'-*\\ 

yl 

1  \ 

\ 

\ 

\ 

B 


M 


D 


Draw  the  necessary  construction  lines  and  show  that  sq.  AH  equals  rect.  AM. 
Finish  the  proof. 

Theorem  26fr.  The  areas  of  two  triangles  having  an  angle  of  one 
equal  to  an  angle  of  the  other  are  to  each  other  as  the  products  of  the 
sides  including  those  angles. 


R 


Given:   AABC  and  &AiXY  with  %.A  = 
AABC      AB-AC 

ToProve: 


Suggestions:  Proof  being  left  to  the  student. 
Place  AABC  in  the  position  of  AAiQR 
AA1RQ_ 

A1X' 


Draw  QY. 


A1R        , 
AlY'  a  a 


AAiYQ~AlY'          AA1XY 

Could  the  case  arise  in  which  QR  intersects  XY  between  X  and  F? 
If  so,  show  that  this  proof  still  holds  or  give  one  that  does  hold. 


AREAS  OF  RECTILINEAR  FIGURES  241 

Theorem  27.  The  area  of  a  trapezoid  is  equal  to  half  the  produc  t 
of  its  altitude  and  the  sum  of  its  bases. 

EXERCISES.    SET  LXXIX.    AREAS 

Numeric 

878.  Find  the  expense  of  paving  a  path  4'  wide  inside  a  square 
piece  of  ground  the  side  of  which  is  50'  if  the  price  is  18  cents  per 
square  yard.  What  would  be  the  cost  if  the  path  were  outside 
the  piece  of  ground? 

a879.*  Find  the  dimensions  of  a  rectangle  given: 

(a)  Area  216  sq.  ft.,  perimeter  60  ft. 

(6)  Area  600  sq.  ft.,  difference  of  the  sides  10  ft. 

(c)  Area  756,  sides  in  the  ratio  -J. 

(d)  Area  340  sq.  ft.,  sum  of  squares  of  two  consecutive  sides 
689  sq.  ft. 

880.  Find  the  change  in  the  area  of  a  triangle  of  base  a  and 
altitude  h  in  the  following  cases : 

(a)  If  a  and  h  are  increased  by  m  and  n,  respectively. 

(6)  If  a  and  h  are  diminished  by  m  and  n,  respectively. 

(c)  If  a  is  increased  by  m,  and  h  diminished  by  n. 

(d)  If  a  is  diminished  by  m,  and  h  increased  by  n. 

881.  A  line  of  division  is  drawn  between  two  sides  of  a  triangle, 
dividing  it  into  a  triangle  and  a  quadrilateral.    What  parts  are 
these  two  figures,  respectively,  of  the  entire  triangle  if  the  line  of 
division  cuts  off  the  following  parts  of  the  two  sides,  reckoned 
from  the  intersection  of  the  sides? 

(a)  |  and  |.  (6)  f  and  f .  (c)  1  and  J. 

(d)  |  and  \.  (e)  £  and  £  (/)   \  and  i. 

882.  Find  the  area  of  a  rhombus,  given: 
(a)  The  diagonals  18  and  12  units. 

(6)  The  sum  of  the  diagonals  12,  and  their  ratio  |-. 

*  As  here,  "a"  precedes  a  problem  which  calls  for  the  solution  of  an  affected 
quadratic  equation. 


242  PLANE  GEOMETRY 

Theoretic 

883.  (a)  Prove,  geometrically,  the  algebraic  formula, 
a(b+c)=ab+ac. 

(b)  Prove,  geometrically,  the  algebraic  formula, 

a(b—  c)=ab  —ac. 

(c)  Prove,  geometrically,  the  algebraic  formula, 


884.  The  area  of  a  rhombus  is  equal  to  half  the  product  of  its 
diagonals. 

885.  If  lines  are  drawn  from  any  point  inside  a  parallelogram  to 
the  four  vertices,  the  sum  of  either  pair  of  triangles  with  parallel 
bases  is  equal  to  the  sum  of  the  other  pair. 

886.  The  accompanying  figures  show  easy  methods  of  trans- 
forming (a)  a  triangle  into  a  parallelogram,  (b)  a  parallelogram  into 
a  triangle,  (c)  a  trapezoid  into  a  parallelogram.    Explain.    Can 
you  give  more  than  one  explanation?    If  so,  upon  what  does  your 
explanation  depend? 


A  F         D 

For  problems  in   construction  based   upon  this   chapter  see 
Chapters  VII  and  VIII. 

TERMS  DEFINED  IN  CHAPTER  H 

Square  on  (a  sect),  square  of  (a  quantity),  medians  of  a  triangle,  rhomboid. 


CHAPTER  III 

SIMILARITY 

The  following  are  terms  with  which  the  student  should  now  be 
familiar: 

Antecedent,  consequent,  extremes,  means,  mean  proportion,  mean 
proportional,  continued  proportion,  inversion,  composition,  division, 
composition  and  division.* 

Theorem  28.  Any  proportion  may  be  transformed  by  alternation. 
Theorem  29.    In  any  proportion  the  terms  may  be  combined  by 
composition  or  division. 

Theorem  30.  In  a  series  of  equal  ratios,  the  ratio  of  the  sum  of 
any  number  of  antecedents  to  the  sum  of  their  consequents  equals 
the  ratio  of  any  antecedent  to  its  consequent. 

Theorem  31.  A  line  parallel  to  one  side  of  a  triangle  divides 
the  other  sides  proportionally. 

Cor.  1.  One  side  of  a  triangle  is  to  either  of  the  sects  cut  off 
by  a  line  parallel  to  a  second  side  as  the  third  side  is  to 
its  homologous  sect. 

COT.  2.    Parallels  cut  off  proportional  sects  on  all  transversals. 
Cor.  3.    Parallels  which  intercept  equal  sects  on  one  transversal 

do  so  on  all  transversals. 
Cor.  4.    A  line  which  bisects  one  side  of  a  triangle,  and  is 

parallel  to  the  second,  bisects  the  third. 
Cor.  6.    A  sect  which  bisects  two 
sides  of  a  triangle  is  par- 
allel to  the  third  side  and 
equal  to  half  of  it. 

Suggestions:   What  means  have  you  for 

proving  lines  parallel? 
How  may  one  sect  be  proved  half  of    „ 

another? 
What  kind  of  figure  would  you  like  to  have? 

*  The  student  will  find  an  alphabetical  index  of  definitions  on  p.  379. 

243 


244  PLANE  GEOMETRY 

The  sect  joining  the  mid-points  of  the  non-parallel  sides  of  a  trape- 
zoid  is  called  its  median. 


f 


Cor.  6.  The  median  of  a  trape- 
zoid  is  parallel  to  the 
bases  and  equal  to  one- 
half  their  sum. 

Suggestions:  Why  draw  a  diagonal? 

Show  that  a  line  through  M  \\Tffi 
will  bisect  AC  and  therefore  DC,  and  hence  coincide  with  MMi. 

Cor.  7.    The  area  of  a  trapezoid  equals  the  product  of  its  median 
and  altitude. 

What  numerical  relation  exists  between  the  median  and  the  bases  of  a 
trapezoid? 

Theorem  32.   A  line  dividing  two  sides  of  a  triangle  proportion- 
ally is  parallel  to  the  third  side. 
Cor.  1.    A  line  dividing  two  sides  of  a  triangle  so  that  these 

sides  bear  the  same  ratio  to  a  pair  of  homologous  sects 

is  parallel  to  the  third  side. 

DIVISION  OF  A  SECT 

A  point  in  a  sect  is  said  to  divide  it  internally,  and  a  point  in  the 
prolongation  of  a  sect  is  said  to  divide  it  externally,  and  in  both  cases 
the  divisions  of  the  sect  are  reckoned  from  one  extremity  to  the 
point  of  division  and  from  that  point  to  the  other  extremity  of  the 
original  sect. 

A _7 B  E 

I  divides  sect  AB  internally  in  the  ratio  =. 

IB 

AE 

E  divides  sect  AB  externally  in  the  ratio  — . 

EB 

NOTE. — When  neither  "internal"  nor  "external"  is  used  to  qualify  the 
division,  internal  is  understood. 

Is  there  any  ratio  into  which  a  sect  cannot  be  divided  externally? 
A  sect  is  said  to  be  divided  harmonically  when  it  is  divided  intern- 
ally and  externally  in  the  same  ratio. 


SIMILARITY 


245 


In  the  foregoing  illustration  AB  is  divided    harmonically  if 
Tl_AE 
IB~EB' 

Is  there  any  ratio  into  which  a  sect  cannot  be  divided 
harmonically? 

Discuss  Theorems  31  and  32  from  the  point  of  view  of  external 
division  of  a  sect. 

Theorem  32cz.  The  bisector  of  an  angle  of  a  triangle  divides 
the  opposite  side  into  sects  which  are  proportional  to  the  adjacent 
sides. 

Given:    AABC;  D  on  AB  and  4.  A  CD  =  %.DCB. 

AD    AC 
To  prove:  55-53. 

Suggestions  :     Compare    A  A  DC    and 

ABDC  in  two  ways.  A  "~*B 

Theorem  32b.  The  bisector  of  an  exterior  angle  of  a  triangle 
divides  the  opposite  side  externally  into  sects  which  are  propor- 
tional to  the  adjacent  sides. 

'TJ  Given:  AABC  with  exterior  $. 
BCD;  EmAB  produced;  %.BCE= 
4.ECD. 


AB    AC 
To  prove:  -  =  - 

Suggestions:    Com- 
pare AAEC  first  with 
ABEC,  and  second  with  ABiEC 

How  should  Bi  be  taken  so  that  ABiEC  may  be  substituted  for  ABECt 
Consider  special  cases  where  a=b,  a>b. 

Cor.  1.    The  bisectors  of  an  adjacent  interior  and   exterior 
angle  of  a  triangle  divide  the  opposite  side  harmonically. 

(Proof  left  to  the  student.    Discuss  special  cases.) 

EXERCISES.    SET  LXXX.    RATIO.    PROPORTION.    PARALLELS 

Numeric 

887.  Find  the  value  of  n  if  (a)  -  =  |,  (6)  -  =  -. 

no        n    c 

216 


246  PLANE  GEOMETRY 


889.  If  find? 

• 


890.  If        =,  find? 

o  O  0 

91.  Form  all  possible  proportions  involving  a,  b,  p,  and  q  if 
ab=pq. 

892.  Find  the  mean  proportionals  between: 

(a)  2  and  50.  (c)   3  and  21.  (e)  a  and  6. 

(b)  2  and  75.  (d)  3  and  19.  (/)  a+6  and  a  -b. 

893.  Find  the  third  proportional  to  (a)  5  and  6,  (6)  a  and  b. 

a    c 

894.  Transform  the  proportion  T=-,  so  that  b  becomes  the  third 

0      u 

term. 


895.  If  -r-  =  ?,  what  is  the  ratio  of  a  to  &? 
o       o 

fi<W?    Tfa-C-e-9     find  a+c+e 

}6'  "   --~'  '  d 


897.  Find  a  fourth  proportional  to  : 

(a)  a,  ab,  c.  (6)  a2,  2a6,  362.  (c)  ««,  x 

898.  Find  a  third  proportional  to: 

(a)  a2&,  a6.  (b)  x*,  2x\  (c)  3«,  Qxy.  (d)  1,  a;. 

899.  Find  mean  proportionals  between  : 

(a)  a2,  62.         (6)  2z3,  Sx.         (c)  I2ax2,  3a3.          (d)  27a263,  36. 

900.  If  a,  bj  c,  be  in  mean  proportion,  show  that: 

(a)  ^r  =  —  •  (6)  (62+6c+c2)  (ac-6c+c2) 

fl~f~D      fl  ~C 

jft^^  *  rr  a     c  xi,  /  \  ab+cd    a2+c2 

d901.*  If  T  =  3,  prove  that:      (a)  -r  —  ^=-5  —  -. 

b    d  ab  —cd    a2  —  c2 


a2+ac+c2 


,                       j  V        U              U,    .    ( 

a  .  o       c  .  a  — |— r         — \— 

— —      -H —  /  x    a     o         c    ( 

P_9.  V—  =  -al 


a  c 


For  guidance  in  proofs  such  as  this  exercise  requires  see  p.  304. 


SIMILARITY  247 

a902.  Solve  the  equations: 


~7    9z+10'  y+3     5y-13     14* 


X    —  &X  ~\~  o      X    —  oX  ~\~  O  f  ,»          £X  —  1 

~~^Z — 5~  =  —QZ — E-  •          W 


Hint:   Transform.    In  doing  so  do  you  lose  any  roots? 

903.  Prove  that  a,  6,  c,  d  are  in  proportion  if 

(a+b  -3c  -3d)  (2a+26  -c+d)  =  (2a+2b  -c  -d)  (a  -b  -3c+3d). 

904.  If  6  is  a  mean  proportional  between  a  and  c,  show  that 
4a2  -962  is  to  462  -9c2  in  the  ratio  of  a2  to  b2. 

905.  If  a,  b,  c,  d  are  in  continued  proportion,  i.e.,  if  7=   =3* 

oca 

prove  that  6-fc  is  a  mean  proportional  between  a+b  and  c+d. 


d906.  If      -  =r}  prove  that  a=c,  or  a 
a+a 


a     /*      P 
d907.  If  -=-=-,  prove  that  each  of  these  ratios  is  equal  to 


908.  Two  numbers  are  in  the  ratio  of  f  ,  and  if  7  be  subtracted 
from  each,  the  remainders  are  in  the  ratio  of  f  ;  find  them. 

909.  What  number  must  be  taken  from  each  term  of  the  ratio 
|f  that  it  may  become  f  ? 

910.  What  number  must  be  added  to  each  term  of  the  ratio 
f  \  that  it  may  become  f? 

911.  If-j-^-  =-£-  =  —  L-   show  that  p+q+r=0. 

b-c    c-a    a-b' 

912.  If  -r^-  =-^-  =  —V,  show  that  x  -y+z=0. 

b+c    c+a    a  -6 


join    rr 
d913.  If  =-  =   = 


ace,  I  a6b-2cbe  +  3aW      ace 

_^_^     show  that  y  6TZ^ 


914.  In  the  accompanying  diagram, 
(a)  If  a  =  3,  6  =  4,  c  =  7,  find  d. 
(6)  If  a  =  5,  c  =  9,  d=10,  find  b. 
(c)   Find   each   sect  in   terms  of  the 
other  three. 


248  PLANE  GEOMETRY 

915.  To  measure  indirectly  the  distance  from  an  accessible 
point  A  to  an  inaccessible  point  B,  run  CK  through  C,  a  point 

from  which  A  and  B  are 
both  visible,  making  <£KCB 
=  ^BCA.  Sight  D  in  line 
with  K  and  C  and  also  in  line 
with  A  and  B.  What  sects 
must  now  be  measured  in 
order  to  compute  A  Bt 

916.    Find   the  area  of   a 
trapezoid,  given  the  median  m,  and  the  altitude  h. 

Theoretic 

917.  The  perpendiculars  dropped  from  the  mid-points  of  two 
sides  of  a  triangle  to  the  third  side  are  equal . 

918.  Lines  joining  the  mid-points 
of   two   opposite   sides  of  a  paral- 
lelogram to  the  ends  of  a  diagonal 
trisect  the  other  diagonal. 

919.  A  line  bisecting  one  of  the 

non-parallel  sides  of  a  trapezoid  and  parallel  to  the  base  bisects 
the  other  non-parallel  side. 

920.  The  sects  joining  the  mid-points  of  the  consecutive  sides 
of  any  quadrilateral  form  a  parallelogram. 

d  Consider  whether  there  are  any  modifications  of  this  fact  in 
the  case  of  (a)  the  parallelogram,  (6)  the  rectangle,  (c)  the  rhombus, 
(d)  the  square. 

d921.  The  mid-points  of  two  opposite  sides  of  a  quadrilateral 
and  the  mid-points  of  the  diagonals  determine  the  vertices  of  a 
parallelogram. 

d922.  The  sects  joining  the  mid-points  of  the  opposite  sides  of  a 
quadrilateral  and  the  sects  joining  the  mid-points  of  the  diagonals 
are  concurrent. 

d923.  If  perpendiculars  are  drawn  from  the  four  vertices  of  a 
parallelogram  to  any  line  outside  the  parallelogram,  the  sum  of  the 
perpendiculars  from  one  pair  of  opposite  vertices  equals  the  sum 
of  those  from  the  other  pair. 

924.  The  triangle  formed  by  two  lines  drawn  from  the  mid-point 
of  either  of  the  non-parallel  sides  of  a  trapezoid  to  the  opposite 
vertices  is  equivalent  to  half  the  trapezoid. 


SIMILARITY 


249 


925.  State  and  prove  the  converse  of  proposition  32a. 

926.  State  and  prove  the  converse  of  proposition  326. 

927.  If  a  sect  PQ  is  divided  harmonically  at  R  and  S,  then  sect 
RS  is  divided  harmonically  at  P  and  Q. 

Construction 

928.  Construct  a  third  proportional  to  two  given  sects. 

929.  Construct  a  fourth  proportional  to  three  given  sects. 

930.  If  a,  6,  c  are  given  sects,  construct  (a)  sect  d,  so  that  r=  , 

»  *.*. 

931.  Divide  a  sect  (a)  internally  into  sects  proportional  to  two 
given  sects,  (b)  externally  into  sects  proportional  to  two  given  sects, 
(c)  harmonically  in  the  ratio  of  two  given  sects. 

932.  Divide  a  given  sect  (a)  internally  in  a  given  ratio  without 
the  use  of  parallels,  (6)  externally,  (c)  harmonically. 

d933.  Construct  two  sects  given  (a)  their  sum  and  their  ratio, 
(b)  their  difference  and  their  ratio. 

d934.  Through  a  given  point  P  draw  a  line  meeting  the^  sides  of 
an  angle  A  in  the  points  B  and  C  so  that  (a)  AB  =  AC,  (b)  UC  =  2AH. 

Theorem  33.  The  homologous  angles  of  similar  triangles  are 
equal,  and  their  homologous  sides  have  a  constant  ratio. 

Theorem  34.  Triangles  are  similar  when  two  angles  of  one  are 
equal,  each  to  each,  to  two  angles  of  another. 

Cor.  1.    Triangles  which  have  their  sides  parallel  or  perpen- 
dicular each  to  each  are  similar. 


Given:   A&A.AB; 
Prove:    AAiBiCi  v>  AABC 
Suggestions :  Show  that : 

(1)  Three  angles  of  A 
one  triangle  cannot  be 
supplementary  to  three 
angles  of  the  other. 

(2)  Two    angles    of 
one  triangle  cannot  be 
supplementary   to   two 
of  the  other. 

(3)  What,  then,  is  the 
fact? 


AB] 


CA. 


250  PLANE  GEOMETRY 

Theorem  35.  Triangles  which  have  two  sides  of  one  propor- 
tional to  two  sides  of  another  and  the  included  angles  equal  are 
similar. 

Theorem  36.  If  the  ratio  of  the  sides  of  one  triangle  to  those  of 
another  is  constant,  the  triangles  are  similar. 

EXERCISES.    SET  LXXXI.    SIMILARITY  OF  TRIANGLES 

Numeric 

935.  If  the  sides  of  a  triangle  are  3,  7,  and  8,  find  the  sides  of  a 
similar  triangle  in  which  the  side  homologous  to  7  is  9. 

936.  If  the  sides  of  a  triangle  are  a,  6,  and  c,  find  the  sides  of  a 
similar  triangle  in  which  the  side  homologous  to  a  is  p. 

Construction 

937.  The  sides  of  a  triangle  are  5,  6,  and  7.    Construct  a  triangle 
similar  to  the  original,  having  the  ratio  of  similitude  3  to  2. 

Q  938.  Construct  a  triangle 

similar  to  the  accompanying 
triangle  with  the  ratio  of 
J2  similitude  equal  to  that  of 
the  two  given  sects  a  and  b. 

Theoretic 

939.  Two  isosceles  triangles  are  similar  if  an  angle  of  one  is 
equal  to  the  homologous  angle  of  the  other. 

940.  Prove  that  the  altitudes  of  a  triangle  are  inversely  propor- 
tional to  the  sides  to  which  they  are  drawn. 

941.  If  the  altitudes  AD  and  BE  in  A  ABC  are  drawn,  prove  that 

AC    DC 

-=—==-;..    Are  the  altitudes  directly  or 


inversely  proportional  to  DC  and  EC1 

942.  If  a  spider,  in  making  its  web, 
makes  A&  \\AB,  B^Ci  \\  BC,  CiA  ||  CD, 
£>!#!  ||  DE,   and  E^  ||  EF,   and  then 
runs  a  line  from  F\  \\  FA,  will  it  strike 
the  point  AI?    Prove  your  answer. 

943.  If  D  is  taken  in  the  leg  A  B  of  an  isosceles  triangle  ABC, 

so  that  CD=AC  (the  base),  then 


SIMILARITY  251 

944.  Isosceles  or  right  triangles  ABC  and  PQR  are  similar  if 


hP     PQ 

945.  The  diagonals  of   a  trapezoid    divide   each    other   pro- 
portionally. 

946.  If  in  triangle  ABC  altitudes  AD  and  BE  meet  at  0,  then: 
(a)  BD  •  DC=DO  •  AD;  (b)  BD  •  AC=BO  -'A5. 

947.  If  in  a  parallelogram  PQRS,  a  sect  Q77  is  drawn  cutting  the 
diagonal  PR  in  V,  the  side  #/S  in  L  and  the  prolongation  of  PS 


948.  In  similar  triangles  homologous  angle  bisectors  are  directly 
proportional  to  the  sides  of  the  triangle. 

949.  In  a  quadrilateral  A  BCD,  right-angled  at  B  and  D,  per- 
pendiculars PE  and  PF  from  any  point  P  in  AC  to  the  sides  BC 

P'E     PF 
and  AD  are  such  that  =-+=-=  1. 

d950.  Every  straight  line  cutting  the  sides  of  a  triangle  (pro- 
duced when  necessary)  determines  upon  the  sides  six  sects,  such 
that  the  product  of  three  non-consecutive  sects  is  equal  to  the 
product  of  the  other  three. 

The  line  XYZ  must  cut  either  (a)  two  sides  of  the  triangle  and 
the  third  side 
produced  (Fig. 
1),  -or  (b)  all 
three  sides  pro- 
duced (Fig.  2). 
The  proof  in 
both  instances 
is  the  same. 

Fig.  1  Fig.  2 

Draw  CD  \\AB.    From  the  similar  triangles 

AX    AZ       .  BY    BX 

— •  = and  ===== 

CD     CZ ,        CY    CD 

.       AX-BY    AZ-BX 

therefore  — : — ==  = — — . 

CY  -CD     CZ  •  CD 

whence  AZ  •  BY  •  CZ=AZ  -BX-CY. 


252 


PLANE  GEOMETRY 


This  theorem  was  discovered  by  Menelaus  of  Alexandria  about 
80  B.C. 

d951.  Prove  the  converse  of  the  last  theorem. 
Let  XY  produced  cut  AC  produced  in  a  point  P. 
Then  AX  •  BY;  CP=AP  -BX-CY. 
But,  by  hypothesis,  A ~X  •  BY  •  CZ=AZ  -  BX  •  CF; 

CP    AP 
whence  ======; 


whence 


or 


AC 

AC 


AZ' 


A  ~ B 

that  is,  P  coincides  with  Z. 

d952.  Lines  drawn  through  the  ver- 
tices of  a  triangle,  and  passing  through 
a  common  point,  determine  upon  the 
sides  six  sects,  the  product  of  three 
non-consecutive  sects  being  equal  to 
the  product  of  the  other  three.  Fig.  2 

The  common  point  0  may  lie  either  inside  or  outside  the  triangle 
(Figs.  1  and  2).  In  both  cases  apply  Ex.  950  to  the  AACD  and 
sect  BOF  and  to  the  ABCD  and  sect  AOE,  then  multiply  the 
results. 


Fig.  1 


This  theorem  was  first  discovered  by  Ceva  of  Milan,  in  1678. 

d953.  Conversely,  if  three  lines  drawn  through  the  vertices  of  a 
triangle  determine  upon  the  sides  six  sects,  such  that  the  product 


SIMILARITY 


253 


of  three  non-consecutive  sects  is  equal  to  the  product  of  the  other 
three,  the  lines  pass  through  the  same  point. 

The  proof  is  similar  to  that  of  Ex.  951. 

Theorem  36a.    The  homologous  angles  of  similar  polygons  are 
equal  ',  and  their  homologous 
sides  have  a  constant  ratio. 
Suggestions:   Why  is  it  that 
the  n-gons  may  be  placed  as  in 
the  accompanying  diagram?   To 
prove  the  second  part  use  the  C 
similar  triangles  thus  formed  and 
show  that  the  ratio  of  the  homo- 
logous sides  is  equal  to  the  ratio 
of  similitude  of  the  polygons. 
Give  the  complete  proof. 

Cor.  1.    If  the  ratio  of  similitude  of  polygons  is  unity,  they  are 

congruent. 

Cor.  2.  The  homologous  diagonals  drawn  from  a  single  vertex 
of  similar  polygons  divide  the  polygons  into  triangles 
similar  each  to  each. 

A  Why    is    A  AiBid   ^  A 

ABC? 

Having  proved  this  can 
you  prove  A  A  \C\D\  v> 
A  A  CD,  and  so  forth? 

Why  is  it  that  only  these 
two  sets  of  A  need  be  proved 
similar? 

Write  the  proof  in  full. 
Theorem  366.     Polygons  whose  homologous  angles  are  equal 
and  whose  homologous 


sides  have  a  constant 
ratio  are  similar. 

Given:  Polygons  ABC. 

N  and 

winch 


JV 


.    AB      BC  = 

Prove:    A1B1C1 

ABC.  .  .N. 


254 


PLANE  GEOMETRY 


PROOF 


Place  A  iBiCi  .  .  .  Ni  as  in  the 
diagram  so  that  A^Bz\\ABt 
and  draw  AA^X  and  BB^Z. 


(2)  If  AB=A1B1   then 


(3)  If  AB>Aj3lt  then 
etc.  and  .ABB^a  is  not  a 

(4)  and  AA2X  and 
at  some  point  O. 

(5)  AAOB 


(6) 
(7) 


intersect 


^ 
AB  ~~BO' 


(8)  and 
(9) 


(10) 

(11)  Draw 

(12)  Then 

(13) 
(14) 
(15) 


C^"  cutting  BBjZ  in  0] 


coin- 


(17) 
cides  with  0. 

(18)  .'.BOi 
through  0. 

(19)  Any  O^L  cutting  sides  of  the 
polygons  in  R  and  L  respectively  is 


BO   and   CC2   passes 


,.   .  ,    ,       tn 
divided  so  that—  = 


(1)  Data. 

(2)  Congruent  polygons  are  similar. 

(3)  The  opposite  sides  of  a  paral- 
lelogram are  equal,  and  conversely. 

(4)  Non-parallel  coplanar  lines  in- 
tersect. 

(5)  Two  angles  equal  each  to  each, 
or  sides  respectively  parallel. 

(6)  Homologous    sides    of    similar 
triangles  have  a  constant  ratio. 

(7)  Corresponding  angles  of  par- 
allels. 

(8)  Data. 

(9)  The  differences  of  equals  less 
equals  are  equal. 

(10)  Corresponding  4?  equal. 

(11)  See  (1),  (2),  (3),  and  (4). 

(12)  See  (5). 

(13)  See  (6). 

(14)  See  (7). 

(15)  Data. 

(16)  Quantities  equal  to  equal  quan- 
tities are  equal  to  each  other. 

(17)  By  division  and  the  products  of 
equals  multiplied  by  equals  are  equal. 

(18)  Two     points     determine     a 
straight  line. 

(19)  See  (6). 


In  the  same  way  it  may  be  proved  that  DD2, .  . .  NN2,  pass  through  0  and 
therefore  AiBiCi.  .  .Ni^ABC.  .  .N  by  the  definition  of  similar  figures. 


SIMILARITY  255 

Cor.  1.  If  homologous  diagonals  drawn  from  a  single  ver- 
tex of  two  polygons  divide  them  into  triangles  similar 
each  to  each  and  similarly  arranged,  the  polygons 
are  similar. 

(Proof  is  left  to  the  student.) 

Theorem  37.  The  perimeters  of  similar  triangles  are  propor- 
tional to  any  two  homologous  sides,  or  any  two  homologous 
altitudes. 

Cor.  1.    Homologous  altitudes  of  similar  triangles  have  the 

same  ratio  as  homologous  sides. 
Cor.  2.    The  perimeters  of  similar  polygons  have  the  same  ratio 

as  any  pair  of  homologous  sides  or  diagonals. 

Apply  propositions  36<z,  37,  and  the  appropriate  law  concernmg  equal 
ratios. 

Theorem  38.  The  areas  of  similar  triangles  compare  as  the 
squares  of  any  two  homologous  sides. 

Cor.  1.  The  areas  of  similar  polygons  compare  as  the  squares 
of  any  two  homologous  sides  or  diagonals. 

Proof  similar  to  that  of  Theorem  37,  Cor.  2.    Give  it  in  detail. 

Cor.  2.  Homologous  sides  or  diagonals  of  similar  polygons 
have  the  same  ratio  as  the  square  roots  of  their 
areas. 

Apply  suitable  axioms  to  the  results  obtained  in  Theorem  38,  Cor.  1. 
Concurrent  lines  are  those  which  pass  through  a  common  point. 

Theorem  38a.  If  two  parallels  are 
cut  by  concurrent  transversals,  the 
ratio  of  homologous  sects  of  the 
parallels  is  constant. 

(Proof  left  to  the  student.) 

Discuss  the  case  where  0  lies  between  the  parallels. 


256 


PLANE  GEOMETRY 


Theorem  385.   //  the  ratio  of  homo- 
logous sects  of  two  parallels  cut  by 
three  or  more  transversals  is  constant, 
the  transversals  are  either  parallel  or 
^concurrent. 

Given:    P  ||  Pt  cut  by  transversals  t,  ti,  h,  in 
X,  Y,  Z  and  Q,  R,  V  respectively,  so  that 


Prove:     t,  tlt  tt  concurrent. 


PROOF 


since 


(1)  If  XY^QR,  t 

Yz 
QRRV 

(2)  If  XY  <  QR^tfh  and  will  inter- 
sect it  at  some  point  O. 


and  similarly 
,  h  ||  «,,  etc. 


««-5-» 

(4)  Suppose  OZ  cuts  PI  in  7,. 


•  -  i-i 

YZ     TZ 

(6)  Then  W=w- 

Complete  the  proof. 


(1)  Why? 

(2)  Why? 

(3)  Why? 

(4)  Why? 

(5)  Why? 

(6)  Why? 


EXERCISES.    SET  J.XXXII.    SIMILARITY  OF  POLYGONS 

Numeric 

954.  The  corresponding  bases  of  two  similar  triangles  are  11  in. 
and  13  in.    The  altitude  of  the  first  is  6  in.    Find  the  corresponding 
altitude  of  the  second. 

955.  The  perimeter  of  an  equilateral  triangle  is  51  in.    Find  the 
side  of  an  equilateral  triangle  of  half  the  altitude. 


SIMILARITY  257 

956.  The  bases  of  a  trapezoid  are  20  in.  and  12  in.,  and  the  alti- 
tude is  4  in.    Find  the  altitudes  of  the  triangles  formed  by  pro- 
ducing the  sides  until  they  meet. 

957.  The  perimeters  of  two  similar  polygons  are  76  and  69. 
If  a  side  of  the  first  polygon  is  4,  find  the  homologous  side  of  the 
other. 

958.  The  sides  of  a  polygon  are  4,  5,  6,  7,  and  8,  respectively. 
Find  the  perimeter  of  a  similar  polygon  if  the  side  corresponding 
to  5  is  7. 

959.  The  sides  of  a  polygon  are  2  in.,  2^-  in.,  3^  in.,  3  in.,  and 
5  in.    Find  the  perimeter  of  a  similar  polygon  whose  longest  side 
is  7  in. 

960.  The  diameter  of  the  moon  is  approximately  3500  miles, 
and  it  is  approximately  250,000  miles  from  the  earth.    At  what 
distance  from  the  eye  will  a  two-inch  disk  exactly  obscure  the  moon? 

Construction 

d961.  Through  a  given  point  PI  draw  a  line  such  that  its  dis- 
tances from  two  given  points  P2  and  P3  shall  be  in  the  ratio  of 
(a)  3  to  5,  (b)  sect  6  to  sect  c. 

d962.  In  the  prolongation  of  the  side  AB  of  triangle  ABC,  find 
point  P  so  that  ZP-]3P  =  CP2. 

963.  Construct  a  triangle  similar  to  a  given  triangle  and  having 
a  given  altitude. 

964.  From  a  given  rectangle  cut  off  a  similar  rectangle  by  a  line 
drawn  parallel  to  one  of  its  sides. 

965.  Construct  a  triangle  having  given: 
(a)  a,  6,  and  the  ratio  of  b  to  c. 

(6)  a,  and  the  ratios  of  a  to  6,  and  a  to  c. 

(c)  a,  and  the  ratios  of  a  to  6,  and  6  to  c. 

(d)  a,  6-j-c,  and  the  ratio  of  b  to  c. 
d(e)  B,  the  ratio  of  a  to  c,  and  he. 

966.  Given  two  sects  AB  and  CD,  and  a  point  P.    Draw  a  line 
XY  through  P — without  producing  AB  and  CD  to  meet — such 
that  AB,  XY,  and  CD  would  be  concurrent  if  produced. 

967.  To  draw  a  parallel  to  one  side  of  a  triangle,  cutting  off 
another  triangle  of  given  perimeter. 

17 


258  PLANE  GEOMETRY 

Theoretic 

968.  The  line  joining  the  mid-points  of  the  bases  of  a  trapezoid 
is  concurrent  with  the  legs  of  the  trapezoid. 

969.  Two  triangles  are  similar  if  an  angle  of  one  is  equal  to  an 
angle  of  the  other  and  the  altitudes  upon  the  including  sides  are 
proportional. 

970.  The  line  bisecting  the  bases  of  a  trapezoid  passes  through 
the  intersection  of  its  diagonals. 

Suggestion:  Prove  that  line  coincident  with  the  line  joining  the  mid-point 
of  one  base  and  the  intersection  of  the  diagonals. 

d971.  Any  sect  drawn  through  the  mid-point  of  one  side  of  a 
triangle  and  limited  by  the  parallel  to  that  side  through  the 
opposite  vertex,  is  divided  harmonically  by  the  second  side  and  the 
prolongation  of  the  third  side. 

972.  If  two  triangles  have  equal  bases  on  one  of  two  parallel 
lines,  and  their  vertices  on  the  other,  the  sides  of  the  triangles 
intercept  equal  sects  on  any  line  parallel  to  these  lines  and  lying 
between  them. 

Reword  the  theorem  (a)  when  one  base  is  half  the  other.     (6) 
When  the  bases  have  any  .given  ratio. 

Theorem  39.  The  altitude  upon  the  hypotenuse  of  a  right  tri- 
angle divides  it  into  triangles  similar  to  each  other  and  to  the 
original. 

Cor.  1.  Each  leg  of  a  right  triangle  is  a  mean  proportional 
between  the  hypotenuse  and  its  projection  upon  the 
hypotenuse. 

Cor.  2.  The  square  of  the  hypotenuse  of  a  right  triangle  is 
equal  to  the  sum  of  the  squares  of  the  other  two 
sides. 

Cor.  3.  The  diagonal  and  the  side  of  a  square  are  incommen- 
surable. 

Cor.  4.  The  altitude  upon  the  hypotenuse  of  a  right  triangle  is 
a  mean  proportional  between  the  sects  it  cuts  off  on  the 
hypotenuse. 

(The  proof  is  left  to  the  student.) 


SIMILARITY 


259 


Theorem  39<z.  In  any  triangle,  the  square  of  the  side  opposite 
an  acute  angle  is  equal  to  the  sum  of  the  squares  of  the  other  two 
sides  diminished  by  twice  the  product  of  one  of  those  sides  by  the 
projection  of  the  other  upon  it. 


b 
FIG.  1. 


I* ^- 


Fio.  2. 


Given:   In  &ABC,  %.A  <90°,  AD^pcb,  the  projection  of  c  upon  b. 
Prove:  a2=62+c2— 2b-p<&. 

PROOF 


(1)  &  BCD  and  ABD  are  rt.  A. 

(2)  .'.  in  ABCD,  a*=hbz+DC\ 

(3)  But  hb2=c2-pcb*in  AABD. 

(4)  and   in   Fig.  1,  DC* '' =  (b-p^ 
while  in  Fig.  2,  DC  =(pcb-b}z  which 
are  identical. 

(5)  /.a'ac' 


(1)  Data  and  def.  of  projection. 

(2)  Why? 

(3)  Why? 

(4)  Why? 


(5)  Why? 


Theorem  39 b.  In  an  obtuse  triangle,  the  square  of  the  side 
opposite  the  obtuse  angle  is  equal  to  the  sum  of  the  squares  of  the 
other  two  sides  increased  by  twice  the  product  of  one  of  those 
sides  by  the  projection  of  the 
other  upon  it. 

Given:   In  &ABC,  3_A>90°,  DA  = 

pcb,  the  projection  of  c  upon  6. 
Prove:  a2  =  62-}~c2-{-2& •/)<.&• 
(Proof  left  to  the  student.) 

Theorem  39c.  7.  The  sum  of  the  squares  of  two  sides  of  a 
triangle  is  equal  to  twice  the  square  of  half  the  third  side,  increased 
by  twice  the  square  of  the  median  to  it. 

This  theorem  is  attributed  to  Appollonius  of  Perga  (c.  225  B.C.). 

77.  The  difference  between  the  squares  of  two  sides  of  a  tri- 
angle is  equal  to  twice  the  product  of  the  third  side  and  the 
projection  of  the  median  upon  it. 


260 


PLANE  GEOMETRY 


Given:    AABC  in  which  a>c,  mb  is  the  median  to  b  and  XD  =p  is  its  projec- 
p  tion  upon  6. 

Prove:  I. 


D 


Outline  of  proof,  which  the  student 
is  asked  to  write. 

(1)  Show  that  $.BDC  is  obtuse 
V*  and  &.BDA  is  acute  by  com- 

paring A  ABD  and  BDC. 

(2)  Use  propositions  39a  and  39&  to  give  the  values  of  a2  and  c2. 

(3)  Combine  these  values  as  indicated  in  I  and  II. 
Consider  this  proposition  when  a=c. 

Cor.  1.    If  mb  represents  the  length  of  the  median  to  side  b  of 
the  triangle  whose  sides  are  a,  b,  c,  then 


Solve  proposition  39c  I  for  mb. 

Theorem  39d.  //  ha  represents  the 
altitude  upon  side  a  of  a  triangle  whose 
sides  are  a,  b,  c,  and  s  represents  its 

semiperimeter,  i.e.,  s  = — '  then 


ha=-nVs(s-a)(s-b)(s-c). 


Fig.  1. 


PROOF 


(1)  At  least  one  of  the  angles  B  or  C  is  acute. 
Suppose  C  is  acute  (FiguresJ_and  2). 
(2) 


(4) 


All  authorities  to  be  given 
by  the  student. 


or 


2a 


2a 


2-c2 \ /c2-(a2  -2ab+b2) \ 

A       2a      ; 


4a2 


FIG.  2. 


SIMILARITY 


261 


3±x/s(s-a)(s-6)(s-c). 
a 

Show  that  this  proof  holds  for  Figures  3  and  4,  i.e.,  when 
C  is  obtuse  or  right. 

Cor.  1.    If  A  stands  for  the  area  of  a  triangle 
whose  sides  are  a,  b,c,  and  whose  semiperimeter  is  s,  then 

A  =  Vs(s-a)(s-b)(s-c) 

By  what  must  the  value  of  ha  be  multiplied  to  obtain  the  value  of  Af 
This  formula  is  known  as  Heron  of  Alexandria's,  to  which  atten- 
tion was  called  on  p.  98,  Ex.  359  of  the  First  Study. 

The  Principle  of  Continuity.  By  considering  both  positive  and 
negative  properties  of  quantities,  a  theorem  may  frequently  be 
stated  so  as  to  include  several  theorems.  For  instance,  Theorems 
39,  Cor.  2,  39a  and  396  may  be  stated  as  a  single  theorem  if  we 
take  into  consideration  the  direction  of  the  projection  of  c  upon  6. 

If  AD,  the  projec- 
tion of  c  upon  6  in 
Fig.  1  be  considered 
positive,  AD  in  Fig. 
3   will  be   negative. 
Therefore    we    may 
say  in  general  that 
a2=62+c2-26.pc6,  for  in  Fig. 
1   that  means  a2=62+c2-26 
(+35),  or  a2^62+c2-26.pc6, 
while  in  Fig.  2  it  means  a2= 
FlG-  3"  62+c2_-  26(0),  or  a2  =  b*  +  c2, 

and  in  Fig.  3  it  meansa2=62+c2  -2b(-DA),  or  a2=62+c2+26-pet. 


262  PLANE  GEOMETRY 

EXERCISES.    SET  LXXXIII.     METRIC  RELATIONS 

Theorems  39a  through  39d  enable  us  to  calculate  the  altitudes 
and  medians  of  a  triangle  in  terms  of  its  sides,  and  the  length  of  the 
projection  of  any  side  upon  any  other,  as  well  as  to  determine 
whether  a  triangle  is  acute,  right  or  obtuse. 

973.  If  in  solving  the  identity  a2=62+c2  -2b-pcb  for  pcb  for  any 
particular  values  of  a,  b,  and  c,  we  find  (a)  pcb  =  0,  then  <£A  =90°, 
(&)  Pc&>0,  then  <£A  <90°,  (c)  pcb  <0,  then  <£A>90°.    Explain. 

974.  Give  the  formulas  for  b  and  c  corresponding  to  those  given 
in  propositions  39a  and  396  for  a. 

975.  From   the   three  formulas   obtained   in   Ex.   974   derive 
formulas  for  pcb,  pab,  pca  or  for  pbc,  pba,  pac- 

976.  Give  a  formula  for  ma,  for  mc,  for  hb,  for  he. 

977.  In  a  right  or  an  obtuse  triangle,  the  greatest  side  is  opposite 
the  right  or  the  obtuse  angle.    Hence  if  a  is  the  greatest  side  of  a 
triangle,  show  that  if  a2>&2-f-c2  the  triangle  is  acute,  if  a2==62+c2 
the  triangle  is  right,  and  if  a2<&2-f  c2  the  triangle  is  obtuse. 

dt978.  Show  that  pcb=c  cos  A  and  hence  that  the  general 
formula  might  be  a2=62+c2  —  2bc  cos  A. 

Theorem  39e.  If  similar  polygons  are  constructed  on  the  sides 
of  a  right  trianglef  as  homologous  sides  f  the  polygon  on  the  hypot- 
enuse is  equal  to  the  sum  of  the  polygons  on  the  other  two  sides. 

If  PI,  PZ,  and  PS  be  similar  polygons  constructed  on  a,  6,  and  c  respect- 

r> 

ively,  as  homologous  sides,  when  <£  C  is  a  right  angle  in   A  ABC,  —  -=  ?, 

PS 

_0 


Give  the  complete  proof. 

EXERCISES.    SET  LXXXIII  (concluded) 
Numeric 

979.  The  base  of  an  isosceles  triangle  is  48  in.    Find  the  altitude 
if  each  arm  equals  50  in. 

980.  Let  ABC  be  a  right  triangle.    The  two  sides  about  the  right 
angle  C  are  respectively  455  and  1,092  feet.     The  hypotenuse 
AB  is  divided  into  two  sects  AE  and  BE  by  the  perpendicular 
upon  it  from  C.    Compute  the  lengths  of  AE,  BE,  and  CE. 


SIMILARITY  263 

981.  (a)  If  two  sides  of  a  triangle  equal  15  and  25,  respectively, 
and  the  projection  of  15  upon  25  equals  9,  what  is  the  value  of  the 
third  side? 

(6)  Is  the  triangle  right,  acute,  or  obtuse? 

982.  The  altitude  of  a  triangle  is  20  in.    A  line  parallel  to  the 
base  and  12  in.  from  the  base  cuts  off  a  triangle  that  is  what  part 
of  the  given  triangle? 

983.  If  the  side  of  an  equilateral  triangle  equals  10  in.,  what 
is  the  length  of  the  projection  of  one  side  upon  another? 

984.  Find  the  projection  of  AB  upon  a  line  XY,  if  AB  and  XY 
include  an  angle  of  45°,  and  AB=2. 

985.  Find  the  side  a  of  the  square  equal  to  an  equilateral  tri- 
angle whose  side  is  s.    Solve  the  equation  for  s  in  terms  of  a. 

986.  Two  sides  of  a  triangle  are  5  and  8,  respectively,  and  include 
an  angle  of  30°.    Find  the  area. 

987.  Find  the  area  of  an  equilateral  triangle  of  which 
(a)  The  side  is  30, 

(6)  The  altitude  is  34, 

(c)  The  side  is  a, 

(d)  The  altitude  is  h. 

988.  Find  the  area  of  a  trapezoid,  given: 

(a)  The  median  ra,  and  altitude  h; 

(b)  The  median  ra,  one  leg  I,  and  the  angle  between  this  and  the 
base  30°; 

(c)  Bases  61  and  62,  and  legs  each  I. 

989.  In  a  trapezoid,  given  the  two  bases  a,  b,  and  the  altitude  h. 
The  legs  are  divided  into  three  equal  parts  by  lines  parallel  to  the 
bases.    Find  in  terms  of  a,  6,  and  h,  the  areas  of  the  three  parts 
into  which  the  trapezoid  is  divided. 

990.  Find  the  side  of  a  rhombus  composed  of  two  equilateral 
triangles  and  equal  to  another  rhombus  whose  diagonals  are  12 
and  18. 

991.  ABC  is  a  triangle  and  AD  the  altitude  upon  BC.     If 
AD  =  13,  and  the  length  of  the  perpendiculars  from  D  to  AB  and 
AC  are  5  and  lOf ,  respectively,  find  the  area  of  the  triangle. 

992.  Find  the  area  of  a  square  in  terms  of  (a)  its  perimeter  p, 
(b)  its  diagonal  d. 


264  PLANE  GEOMETRY 

993.  Find  the  dimensions  of  a  rectangle  given: 

(a)  Its  perimeter  p  and  area  a : 

(b)  Its  length  I  and  diagonal  d; 

(c)  Its  diagonal  d  and  the  ratio  of  its  length  to  its  width  r. 

994.  Find  the  projection  of  AB  upon  XY,  if  AB  =  m,  and  the 
two  lines  include  an  angle  (a)  of  60°,  (6)  of  30°. 

995.  In  triangle  abc,  a  =  8,  6  =  15,  and  the  angle  opposite  c  equals 
60°.    Findc. 

996.  In  triangle  abc,  a=»3,  6  =  5,  and  the  angle  opposite  c  equals 
120°.    Findc. 

997.  In  triangle  abc,  a  =  7,  6  =  8,  and  the  angle  opposite  c  equals 
120°.     Find  c. 

998.  Two  side.s  of  a  triangle  are  20  and  30,  respectively,  and 
include  an  angle  of  45°.    Find  the  third  side. 

999.  Two  sides  of  a  triangle  are  16  and  12  in.,  respectively,  and 
include  an  angle  of  60°.    Find  the  third  side. 

1000.  Find  the  area  of  a  rectangle  in  terms  of  its  length  I,  and 
diagpnal  d. 

1001.  In  triangle  abc}  *a  =  20,  6  =  15,  and  c  =  7.    Find  the  projec- 
tion of  6  upon  c.    Is  the  triangle  obtuse  or  acute? 

1002.  In  a  quadrilateral  ABCD,  AB  =  10,  BC  =  17,  CD  =  13, 
DA  =20,  and  AC  =  21.    Find  the  diagonal  BD. 

1003.  Two  sides  of  a  triangle  are  17  and  10;  the  altitude  upon  the 
third  side  is  8.     What  is  the  length  of  the  third  side? 

1004.  The  sides  of  a  triangle  are  7,  8,  and  9,  respectively.    Find 
the  length  of  the  median  to  8. 

1005.  The  sides  of  a  triangle  are  10,  5,  and  9,  respectively. 
Find  the  length  of  the  median  to  9. 

1006.  The  sides  of  a  triangle  are  22,  20,  and  18,  respectively. 
Find  the  length  of  the  median  to  18. 

1007.  The  sides  of  a  triangle  are  9,  10,  and  17,  respectively. 
Find  the  three  altitudes. 

1008.  The  sides  of  a  triangle  are  11,  25,  and  30,  respectively. 
Find  the  three  altitudes. 

1009.  The  sides  of  a  triangle  are  12,  14,  and  15   respectively. 
Find  the  three  altitudes. 

1010.  (a)  Find  the  altitude  of  an  equilateral  triangle  with  side  s. 
(6)  Find  the  side  of  an  equilateral  triangle  with  altitude  h. 


SIMILARITY  265 

1011.  Find  the  area  of  a  triangle  whose  sides  are  respectively 
(a)  13,  14,  15,  (6)  9,  10,  17,  (c)  11,  25,  30. 

d!012.  The  sides  of  a  triangle  are  as  8  to  15  to  17.  Find  the 
altitudes  if  the  area  is  480  sq.  ft. 

1013.  Find  one  diagonal  of  a  parallelogram,  given  the  sides 
a,  6,  and  the  other  diagonal  g. 

Construction 

1014.  Given  any  sect  as  unit,  construct  a  sect  which  is  -\/2, 

\/3>  \/5  units. 

PA      1 

1015.  In  a  given  sect  AB  find  a  point  P  such  that  (a)  -^-5  =  —  ^, 

"&     V2 

PA         1 


1016.  Construct  an  equilateral  triangle  equal  to  (a)  the  sum  of 
two  given  equilateral  triangles,  (6)  their  difference. 

1017.  Construct  a  polygon  similar  and  equal  to  (a)  the  sum  of 
two  given  similar  polygons,  (&)  their  difference. 

1018.  Construct  a  square  equal  to  the  sum  of  3,  4,  5  given 
squares. 

(For  other  construction  problems  based  on  this  character,  see 

Chapter  VII.) 

Theoretic 

1019.  The  median  drawn  from  the  extremities  of  the  hypot- 
enuse of  the  right  triangle  ABC  are  BE,  CF\  prove  that 


d!020.  In  a  certain  triangle  ABC,  AC2  -BC2=^AB*\  show 
that  a  perpendicular  dropped  from  C  upon  AB  will  divide  the 
latter  into  sects  which  are  to  each  other  as  3  to  1. 

1021.  If  ABC  is  a  right  triangle,  C  the  vertex  of  the  right  angle, 
D  any  point  in  AC,  then  BD2+AC2^AB2+DC2. 

1022.  The  sum  of  the  squares  of  the  four  sides  of  a  parallelogram 
is  equal  to  the  sum  of  the  squares  of  its  diagonals. 

_10231_If  Jn  the  parallelogram  ABCD   £4=60°,  ZC2=ZB2+ 
BC2+AB-BC. 

1024.  The  sum  of  the  squares  of  the  medians  of  a  triangle  is 
three-fourths  the  sum  of  the  squares  of  its  sides. 


266  PLANE  GEOMETRY 

dl025.  The  sum  of  the  square  on  the  difference  of  the  legs  of  a 
right  triangle,  and  twice  the  rectangle  whose  sides  are  the  legs  of 
the  triangle,  is  equal  to  the  square  on  the  hypotenuse. 

State  this  exercise  in  algebraic  form  and  prove  it. 

1026.  One-half  the  sum  of  the  squares  on  the  sum  and  difference 
of  the  legs  of  a  right  triangle  is  equal  to  the  square  on  the 
hypotenuse  . 

State  and  prove  this  exercise  algebraically. 

dl027.  Two  similar  parallelograms  are  to  each  other  as  the  pro- 
ducts of  their  diagonals. 

1028.  If  in  triangle  ABC,  AB=BC  and  altitudes  AD  and  BE 

•  A  i  s\  ji       .oC 

intersect  at  0,  then  TT»= 


1029.  If  in  a  triangle  the  ratio  of  the  squares  of  two  sides  is 
equal  to  the  ratio  of  their  projections  upon  the  third  side,  the 
triangle  is  a  right  triangle. 

dlOSO.  The  sum  of  the  squares  of  the  sides  of  a  quadrilateral 
is  equal  to  the  sum  of  the  squares  of  the  diagonals  increased  by 
four  times  the  square  of  the  sect  joining  their  mid-points. 

dlOSl.  If  perpendiculars  are  drawn  to  the  sides  of  a  triangle 
from  any  point  within  it,  the  sum  of  the  squares  of  three  alternate 
sects  cut  off  on  the  sides  is  equal  to  the  sum  of  the  squares  of  the 
three  remaining  sects. 

1032.  If  ABCD  is  a  sect_such  that  AB=BC==C^D,  and  P  is 
any  other  point,  prove  that  PA2+3PC2=PD2-\-3PB*. 


CHAPTER  IV 

LOCUS 

Theorem  40.    The  locus  of  points  equidistant  from  the  ends  of 
a  sect  is  the  perpendicular  bisector  of  the  sect. 

Cor.  1.    Two  points  equidistant  from  the  ends  of  a  sect  fix  its 
perpendicular  bisector. 

Theorem  41.    The  locus  of  points  equidistant  from  the  sides  of 
an  angle  is  the  bisector  of  the  angle. 
Cor.  1.    The  locus  of  points  equidistant  from  two  intersecting 

lines  is  a  pair  of  lines  bisecting  the  angles. 
(Concurrent  lines  are  those  which  pass  through  a  common  point.) 

Theorem  41a.    The  bisectors  of  the  angles  of  a  triangle  are 
concurrent  in  a  point  equidistant  from  the  sides  of  the  triangle. 


Given:   &ABC;  ^BAZ^^ZAC,  %_ACY=4.YCB, 

To  prove:  AZ,  CY,  and  BX  are  concurrent  in  a  point  equidistant  from 

the  sides. 

Suggestions  for  proof:  If  BX  were  parallel  to  CY  what  relation  would  exist 
between  ^YCB  and  ^.CBX? 

If  0  is  a  point  in  YC,  how  is  it  located  with  regard  to  BC  and  ACT 
If  O  is  a  point  in  BX,  how  is  it  located  with  regard  to  BC  and  ABf 
Why,  then,  must  O  be  on  AZ? 

287 


PLANE  GEOMETRY 


0 

X 

X 

V/2 

M          c/ 

/2             A 

Theorem  416.  The  perpendicular  bisectors  of  the  sides  of 
a  triangle  are  concurrent  in  a  point  equidistant  from  the 
vertices. 

Giveni^AABC;  MM^AMB,  RR, 
±ARC,    PPiA.CPB;   AM  = 
MB,AR=RC;  CP=PB. 
To  prove:   MMi,   PPi,  RRi    are 
concurrent  in  a  point   equi- 
distant from  A,  B,  and  C. 
Suggestions  for   proof:    If  MMi 
were    parallel  to  PP\   what 
relation  would  exist  between 
MMi  and  CB1 

Therefore,    what     relation 
would  exist  between  CB  and  BA? 

If  0  is  a  point  in  MMi  how  is  it  located  with  regard  to  A  and  5? 
If  O  is  a  point  in  PPi  how  is  it  located  with  regard  to  C  and  5? 
Why,  then,  is  0  a  point  in  RRJ 

Theorem  41c.    The  altitudes  of  a  triangle  are  concurrent. 
Given: AAJ5C;     CH  A. 

AHB,    BTA.  CTA, 

ALA.  CLB. 
To  prove:    AL,  BT,  CH 

are  concurrent. 
Analysis  of  proof:  If  AL, 

BT,    CH    were     the 

perpendicular    bisec-  A  ^-JI 

tors  of   the  sides  of 

AAiBiCi  they  would  be  concurrent. 
If  they  are  to  be  such,  how  should  the  sides 

of  AAiBiCi  be  drawn  through  A,  B,  and  C  to 

make  AL_\_BiCi,  BT±Aidt  and  CH  A.A1B11 
If  ACi  =ABlf  BCi=BAi,  and  CAl  =CBi,  then 

this  construction  would  make  possible  a  proof. 

Give  a  synthetic  proof.* 

NOTE. — Where  an  obtuse  triangle  is  involved,  show  that  no  separate 
proof  is  necessary. 


H 


*  For  notes  on  various  types  of  proof  see  Chapter  VI,  p.  297. 


LOCUS 


269 


Theorem  41d.    The  medians  of  a  triangle  are  concurrent  in  a 
point  of  trisection  of  each. 

Given :  A  ABC;  BM = MC,  B 


M  in  BC,  MI 

in  BA,  M2  in  AC.  . 
To  prove:  AM,  BM2,  CM! 
meet   in   a   point    O 
such  that  Ad=2MO, 


2M1O. 

Notes  on  proof:  Why  can- 
not CM\  be  parallel 


Bisect  AO  at  X  and  CO  at  F. 
How  is  JTF  related  to  AC?    (Consider  AAOC.) 
How  is  MMi  related  to  AC?    (Consider  AABC.) 
Therefore,  how  is  MMi  related  to  XYt 

Prove  AXOY^  AMOM1}  and  hence  XO=OM  and  MiO=OY. 
Therefore,  CO=2MiO  and  AO=2MO. 

Consider,  now,  medians  AM  and  BMi.    Can  they  be  parallel,  and,  if 
not,  why  would  their  point  of  intersection  Oi  coincide  with  0? 

EXERCISES.    SET  LXXXIV.    LOCUS  * 

1033.  Find  the  locus  of  the  mid-points  of  sects  drawn  from  a 
common  point  to  a  given  line. 

1034.  Find  the  locus  of  the  points  in  which  the  sects  mentioned 
in  Ex.  1033  are  divided  in  the  ratio  5  to  8. 

1035.  Find  the  locus  of  the  points  in  which  the  sects  mentioned 
in  Ex.  1033  are  divided  in  the  ratio  of  two  given  sects  a  and  b. 

1036.  Find  the  locus  of  the  mid-points  of  sects  connecting  points 
on  two  parallels. 

1037.  Lines  are  drawn  parallel  to  one  side  of  a  triangle  and  are 
terminated  by  the  other  two  sides.    What  is  the  locus  of  their 
mid-points? 

*  Various  terms  are  used  in  stating  locus  exercises.  We  shall  follow  the 
most  usual  interpretations,  which  are:  (1)  No  locus  exercise  need  be  proved 
unless  a  proof  is  definitely  called  for,  indicated  by  "prove";  (2)  an  accurate 
construction  is  called  for  when  the  terms  "plot"  or  "construct"  are  used; 
(3)  the  terms  "describe"  or  "find"  are  used  in  calling  for  a  statement  of  what 
the  locus  is. 


270  PLANE  GEOMETRY 

1038.  Parallel  sects  are  drawn  with  their  extremities  in  the  sides 
of  an  angle.    Find  the  locus  of  their  mid-points. 

1039.  What  is  the  locus  of  the  vertices  of  triangles  having  (a)  a 
given  base  and  a  given  altitude?   (6)  a  given  base  and  a  given  area? 

d!040.  Find  a  point  within  a  triangle  such  that  the  lines  joining 
it  to  the  vertices  shall  divide  the  triangle  into  three  equal  parts. 

1041.  If  AB  be  a  fixed  sect,  find  the  locus  of  a  point  which  moves 
so  that  its  distance  from  the  nearest  point  in  AB  is  always  equal 
to  a  given  sect  c. 

How  does  this  locus  differ  from  the  one  obtained  if  for  the  word 
"  sect  "  we  substitute  "  line  "? 

1042.  If  PQRS  be  a  rhombus,  such  that  Q  and  S  lie  on  two  fixed 
lines  through  P,  find  the  locus  of  R. 

1043.  If  PQRS  be  a  parallelogram  of  constant  area  and  given 
base  PS,  find  the  loci  of  R  and  Q.__ 

1044.  If  A  be  a  fixed  point,  BC  a  fixed  line,  n  any  integral 
number,  P  any  point  in  BC,  and  Q  a  point  in  AP  or  PA  produced 

so  that  AQ=n-AP,  find  the  locus  of  Q. 

1045.  Find  the  locus  in  the  last  exercise  if  AP=n-AQ. 

1046.  If  in  the  APQR  a  sect  QS  be  drawn  to  any  point  in  the 

QT 

base,  find  the  locus  of  a  point  T  on  this  sect  such  that  the  ratio  — — ; 

1  o 
is  constant. 

Justify  the  two  expressions  "  the  locus  of  points  "  and  "  the 
locus  of  a  point." 

1047.  If  from  the  intersection  of  the  diagonals  of  a  parallelo- 
gram sects  are  drawn  to  the  perimeter,  find  the  locus  of  the  point 
in  these  sects  such  that  the  ratio  of  the  parts  into  which  the  sect 

fn 
is  divided  is  (a)  constant,  (6)  equal  to  a  given  ratio  --  or  (c)  equal 

ft/ 

to  the  ratio  of  two  given  sects  a  and  6. 

1048.  Given  a  square  with  side  3  in.    Construct  the  locus  of  a 
point  P  such  that  the  distance  from  P  to  the  nearest  point  of  the 
square  is  1  in. 

1049.  Upon  a  given  base  is  constructed  a  triangle,  one  of  whose 
base  angles  is  double  the  other.    The  bisector  of  the  larger  base 
angle  meets  the  opposite  side  at  the  point  P.    Find  the  locus  of  P. 


LOCUS  -  271 

dl050.  What  is  the  locus  of  points,  the  distances  of  which 
from  two  intersecting  lines  are  to  each  other  as  m  to  ft? 

dl051.  Find  the  locus  of  points  the  sum  of  whose  distances  from 
two  given  parallel  lines  is  equal  to  a  given  length.  Discuss  all 
possible  cases. 

d!052.  Find  the  locus  of  points  the  difference  of  whose  distances 
from  two  given  parallel  lines  is  equal  to  a  given  length.  Discuss. 

dl053.  Find  the  locus  of  points  the  sum  of  whose  distances  from 
two  given  intersecting  lines  is  equal  to  a  given  length. 

d!054.  Find  the  locus  of  points  the  difference  of  whose  distances 
from  two  given  intersecting  lines  is  equal  to  a  given  length. 

1055.  The  vertex  A  of  a  rectangle  ABCD  is  fixed,  and  the  direc- 
tion of  the  sides  AB  and  AD  also  are  fixed.  Plot  the  locus  of  the 
vertex  C  if  the  area  of  the  rectangle  is  constant. 

d!056.  Plot  the  locus  of  a  point  if  the  product  of  its  distances 
from  two  perpendicular  lines  is  constant. 

d!057.  Plot  the  locus  of  a  point  P  such  that  the  sum  of  the 
squares  of  its  distances  from  two  fixed  points  is  constant. 

d!058.  Plot  the  locus  of  a  point  such  that  the  difference  of  the 
squares  of  its  distances  from  two  fixed  points  is  constant. 

dl059.  Given  the  base  of  a  triangle  in  magnitude  and  position 
and  the  difference  of  the  squares  of  the  other  two  sides,  plot  the 
locus  of  the  vertex. 

1060.  Given  a  square  ABCD.  Let  E  be  the  mid-point  of  CD, 
and  draw  BE.  A  line  is  drawn  parallel  to  BE  and  cutting  the 
square.  Let  P  be  the  mid-point  of  the  sect  of  this  line  within  the 
square.  Construct  the  locus  of  P  as  the  line  moves,  always  remain- 
ing parallel  to  BE. 

Other  locus  exercises  will  be  found  in  the  chapter  on  "Circles," 
pp.  273,  275,  276,  283,  284,  288,  as  well  as  in  the  chapter  on 
"  Methods  of  Attacking  Problems,"  p.  306,  et  seq. 


CHAPTER  V 

THE  CIRCLE 
Theorem  42.    Three  points  not  in  a  straight  line  fix.  a  circle. 

Theorem  43.    In  equal  circles,  equal  central  angles  intercept 
equal  arcs,*  and  conversely. 

Theorem  43a.    In  equal  circles,  the  greater  of  two  central  angles 
intercepts  the  greater  arc,  and  conversely. 

Suggestion:  Lay  off  the  smaller  central  angle  on  the  greater  to  prove  the  direct. 
What  may  be  done  in  the  case  of  the  converse? 

Theorem  44.    In  equal  circles,  equal  arcs  are  subtended  by 
equal  chords,  and  conversely. 

Theorem  44a.    In  equal  circles,  unequal  arcs  are  subtended  by 
chords  unequal  in  the  same  order,  and  conversely. 

Suggestions:   If  radii  are  drawn,  what  do  we  know  of  the  triangles  formed? 
Then  what  method  of  proving  sects  unequal  may  be  used  in  the  proof  of 

the  direct? 
In  the  proof  of  the  converse,  what  is  the  only  method  you  are  ready  to 

use  in  order  to  prove  arcs  unequal? 
Write  the  proofs  of  both  parts  of  this  theorem. 

Theorem  45.    A  diameter  perpendicular  to  a  chord  bisects  it 
and  its  subtended  arcs. 

Cor.  1.    A  radius  which  bisects  a  chord  is  perpendicular  to  it. 
Cor.  2.    The  perpendicular  bisector  of  a  chord  passes  through 
the  center  of  the  circle. 

Theorem  46.    In  equal  circles,  equal  chords  are  equidistant 
from  the  center,  and  conversely. 

Theorem  46a.    In  equal  circles  the  distances  of  unequal  chords 
from  the  center  are  unequal  in  the  opposite  order,  and  conversely. 

*  Such  arcs  are  actually  congruent,  but  we  are  following  custom  in  using 
the  word  "equal." 
272 


THE  CIRCLE 


273 


Axioms  of  Inequal- 
ity (continued).  5. 
Squares  of  positive  un- 
equals  are  unequal  in 
the  same  order.  Illus- 
trate. 
Given:  OCaQd;  chord 


AB>  chord  DE;  CY  LAYB,  CiX  ±DXE. 
Prove:   CY<dX. 

PROOF  OF  THE  DIRECT 

Authorities  left  for  the  student  to 
supply. 


But  AB>DE,  .'.  YB>XE; 


%aadv. 

Can  the  same  method  be  used  for  the  proof  of  the  converse? 
Give  the  proof  in  full. 

NOTE. — Why  is  it  better  to  use  a  direct  method  rather  than  the  method 
of  exclusion  in  the  proof  of  the  converse? 

Theorem  47.    A  line  perpendicular  to  a  radius  at  its  outer  ex- 
tremity is  tangent  to  the  circle. 
Cor.  1.    A  tangent  to  a  circle  is  perpendicular  to  the  radius 

drawn  to  the  point  of  contact. 
Cor.  2.    The  perpendicular  to  a  tangent  at  the  point  of  contact 

passes  through  the  center  of  the  circle. 
Cor.  3.    A  radius  perpendicular  to  a  tangent  passes  through  the 

point  of  contact. 
Cor.  4.    Only  one  tangent  can  be  drawn  to  a  circle  at  a  given 

point  on  the  circle. 

Theorem  48.    Sects  of  tangents  from  the  same  point  to  a  circle 
are  equal. 

Theorem  49.    The  line  of  centers  of  two  tangent  circles  passes 
through  their  point  of  contact. 

Theorem  49a.    The  line  of  centers  of  two  intersecting  circles 
is  the  perpendicular  bisector  of  their  common  chord. 

What  is  the  locus  of  points  equally  distant  from  the  ends  of  a  sect?  Where, 
then,  do  the  centers  of  these  circles  lie? 
18 


274  PLANE  GEOMETRY 

EXERCISES.     SET  LXXXV.     THE  STRAIGHT  LINE  AND  THE 

CIRCLE 

1061.  What  methods  can  you  now  add  to  those  known  before 
this  chapter  of  showing: 

(a)  Sects  equal?     (6)  Angles  equal?     (c)  Sects  unequal?     (d) 
Angles  unequal?     (e)  Sects  perpendicular? 

1062.  Can  you  now  mention  certain  relations  of  a  new  kind  of 
element?    If  so,  what  are  they? 

Numeric 

1*063.  Two  parallel  chords  of  a  circle  are  4  and  8  units  in  length, 
and  their  distance  apart  is  3  units.    What  is  the  radius? 

1064.  Two  parallel  chords  of  a  circle  are  d  and  k  in  length,  and 
their  distance  apart  is  /.    What  is  the  radius? 

1065.  Find  the  length  of  a  tangent  from  a  point  15"  from  the 
center  of  a  circle  whose  radius  is  5". 

1066.  Find  the  radius  of  a  circle  if  the  length  of  a  tangent  from 
a  point  23"  from  the  center  is  16". 

1067.  Find  the  length  of  the  longest  chord  and  of  the  shortest 
chord  that  can  be  drawn  through  a  point  1'  from  the  center  of  a 
circle  whose  radius  is  20". 

1068.  The  radius  of  a  circle  is  5".    Through  a  point  3"  from 
the  center  a  diameter  is  drawn,  and  also  a  chord  perpendicular  to 
the  diameter.    Find  the  length  of  this  chord,  and  the  distance  (to 
two  decimal  places)  from  one  end  of  the  chord  to  the  ends  of  the 
diameter. 

1069.  The  span  (chord)  of  a  bridge  in  the  form  of  a  circular  arc 
is  120',  and  the  highest  point  of  the  arch  is  15'  above  the  piers. 
Find  the  radius  of  the  arc. 

1070.  The  line  of  centers  of  two  circles  is  30.    Find  the  length 
of  the  common  chord  if  the  radii  are  8  and  26  respectively. 

1071.  Two  circles  touch  each  other,  and  their  centers  are  8/r 
apart.    The  radius  of  one  of  the  circles  is  5".     What  is  the  radius 
of  the  other?    (Two  solutions.) 

1072.  If  the  radii  of  two  concentric  circles  are  denoted  by  a  and 
b,  respectively,  find  the  radius  of  a  third  circle  which  shall  touch 
both  given  circles  and  contain  the  smaller. 


THE  CIRCLE 


275 


Locus 

1073.  Find  the  locus  of  the  center  of  a  circle  which  has  a  given 
radius  and  is  tangent  to  a  given  circle. 

1074.  Find  the  locus  of  the  extremity  of  a  tangent  of  given  length 
drawn  to  a  given  circle. 

1075.  Two  equal  circles  are  tangent  to  each  other  externally. 
Find  the  locus  of  the  centers  of  all  circles  tangent  to  both. 

1076.  A  sect  so  moves  that  it  re- 
mains parallel  to  a  given  line,  and 
so  that  one  end  lies  on  a  given  cir- 
cle. Find  the  locus  of  the  other  end. 
Does  the    accompanying  diagram 
give  the  complete  locus? 

1077.  What  is  the  locus  of  the 

mid-points  of  parallel  chords  of  a  circle?    Prove  the  correctness  of 
your  statement. 

1078.  From  a  point  outside  how  many  tangents  are  there  to  a 

circle?    Prove. 

1079.  Find   the  locus   of  the 
mid-point  of  a  sect  that  is  drawn 
from  a  given  external  point  to  a 
given  circle. 

1080.  A  straight  line  3  in.  long 
moves  with  its  extremities  on  the 

perimeter  of  a  square  whose  sides  are  4  in.  long.    Construct  the 
locus  of  the  mid-point  of  the  moving  line. 

1081.  A  circular  basin  16  in.  in  diameter  is  full  of  water,  and 
upon  the  surface  there  floats  a  thin  straight  stick  1  ft.  long.  Shade 
that  region  of  the  surface  which  is  inaccessible  to  the  mid-point 
of  the  stick,  and  describe  accurately  its  boundary. 

1082.  The  image  of  a  point  in  a  mirror  is  apparently  as  far 
behind  the  mirror  as  the  point  itself  is  in  front.    If  a  mirror  revolves 
about  a  vertical  axis,  what  will  be  the  locus  of  the  apparent  image 
of  a  fixed  point  1  ft.  from  the  axis? 

1083.  In  the  rectangle  A  BCD  the  side  AB  is  twice  as  long  as  the 
side  BC.    A  point  E  is  taken  on  the  side  AB,  and  a  circle  is  drawn 


276  PLANE  GEOMETRY 

through  the  points  C,  D,  and  E.    Plot  the  path  of  the  center  of  the 
circle  as  E  moves  from  A  to  B. 

1084.  Find  the  locus  of  a  point  P  such  that  the  ratio  of  its  dis- 
tances from  two  fixed  points  is  equal  to  the  constant  ratio  m  to  n. 

Construction  * 

1085.  Find  the  center  of  a  given  circle. 

1086.  Inscribe  a  circle  in  a  given  triangle. 

1087.  Circumscribe  a  circle  about  a  given  triangle. 

1088.  Escribe  circles  about  a  given  triangle.      (See  p.  359.) 

1089.  Through  a  given  point  in  a  circle  draw  the  shortest  pos- 
sible chord. 

1090.  Inscribe  a  circle  in  a  given  sector.    VII. 

1091.  With  its  center  in  a  given  line  construct  a  circle  which 
shall  be: 

(a)  Tangent  to  another  given  line  at  a  given  point. 

(6)  Tangent  to  two  other  given  lines.f 

(c)  Tangent  at  a  given  point  to  a  given  circle.    VII. 

1092.  Construct  a  circle  of  given  radius  r,  which  shall : 

(a)  Pass  through  a  given  point  and  be  tangent  to  a  given  line; 

(b)  Pass  through  a  given  point  and  be  tangent  to  a  given  circle; 

(c)  Be  tangent  to  a  given  line  and  a  given  circle; 

(d)  Be  tangent  to  two  given  circles.    VII. 

1093.  Construct  a  circle  tangent  to  two  given  lines  and  having 
its  center  on  a  given  circle.    VII. 

1094.  An  equilateral  triangle  ABC  is  2  in.  on  a  side.    Construct 
a  circle  which  shall  be  tangent  to  A  B  at  the  point  A  and  shall  pass 
through  the  point  C.    VII. 

1095.  To  a  given  circle  draw  a  tangent  that  shall  be  parallel 
to  a  given  line. 

1096.  Draw  two  lines  making  an  angle  of  60°,  and  construct  all 
the  circles  of  J^  in.  radius  that  are  tangent  to  both  lines. 

*  While  some  more  or  less  difficult  construction  problems  have  been  inserted 
at  this  point,  they  have  been  primarily  inserted  for  the  benefit  of  those  pupils 
who  wish  to  test  their  power,  and  when  found  too  difficult  may  well  be  omitted 
until  Chapter  VII  has  been  studied.  Such  problems  will  be  followed  by  the 
Roman  number  "VII." 

t  See  p.  311  for  the  section  dealing  with  "The  Discussion  of  a  Problem." 


THE  CIRCLE  277 

dl097.  Construct  an  equilateral  triangle, 
having  given  the  radius  of  the  circumscribed 
circle.  VII. 

1098.  Construct  a  circle,  touching  a  given 
circle  at  a  given  point,  and  touching  a  given 
line.    VII. 

1099.  In  a  given  square  inscribe  four  equal 

circles,  so  that  each  shall  be  tangent  to  two  of  the  others,  and  also 
tangent  to  two  sides  of  the  square. 

dllOO.  In  a  given  square  inscribe  four  equal  circles,  so  that  each 
shall  be  tangent  to  two  of  the  others,  and  also  tangent  to  one  and 
only  one  side  of  the  square.  VII. 

dllOl.  In  a  given  equilateral  triangle  inscribe  three  equal  circles 
tangent  each  to  the  other  two,  each  circle  being  tangent  to  two 
sides  of  the  triangle. 

1102.  Draw  a  tangent  to  a  given  circle  such  that  the  sect  inter- 
cepted between  the  point  of  contact  and  a  given  line  shall  have  a 
given  length.    VII. 

Theoretic 

1103.  If  two  chords  intersect  and  make  equal  angles  with  the 
diameter  through  their  point  of  intersection,  they  are  equal. 

1104.  The  area  of  a  circumscribed  polygon  is  equal  to  half  the 
product  of  its  perimeter  by  the  radius  of  the  inscribed  circle. 

1105.  If  two  common  external  tangents  or  two  common  internal 
tangents  are  drawn  to  two  circles,  the  sects  intercepted  between  the 
points  of  contact  are  equal. 

1106.  If  two  circles  are  tangent  externally,  the  common  internal 
tangent  bisects  the  two  common  external  tangents. 

1107.  A  line  tangent  to  two  equal  circles  is  either  parallel  to  the 
sect  joining  their  centers  or  bisects  it. 

1108.  A  coin  is  placed  on  the  table.   How  many  coins  of  the  same 
denomination  can  be  placed  around  it,  each  tangent  to  it  and 

tangent    to    two    of    the   others? 
Prove  your  answer. 

1109.  If  through  any  point  in  the 
convex  arc  included  between  two 
tangents  a  third  tangent  is  drawn, 
a  triangle  will  be  formed,  the  peri- 


278  PLANE  GEOMETRY 

meter  of  which  is  constant  and  equal  to  the  sum  of  the  two 
tangents. 

1110.  If  a  triangle  is  inscribed  in  a  triangle  ABC,  whose  semi- 
perimeter  is  s,  the  sects  of  its  sides  from  the  vertices  to  the  points 
of  contact  are  equal  to  s  —a,  s  —b,  and  s  — c. 

1111.  The  perimeter  of  an  inscribed  equilateral  triangle  is  equal 
to  half  the  perimeter  of  the  circumscribed  equilateral  triangle. 

1112.  The  radius  of  the  circle  inscribed  in  an  equilateral  tri- 
angle is  equal  to  one-third  of  the  altitude  of  the  triangle. 

dl!13.  In  a  circumscribed  quadrilateral  the  sum  of  two  opposite 
sides  is  equal  to  the  sum  of  the  other  two  sides,  and  a  circle  can  be 
inscribed  in  a  quadrilateral  if  the  sum  of  two  opposite  sides  is 
equal  to  the  sum  of  the  other  two  sides. 

dl!14.  In  what  kinds  of  parallelograms  can  a  circle  be  inscribed? 
Prove. 

1115.  The  diameter  of  the  circle  inscribed  in  a  right  triangle 
is  equal  to  the  difference  between  the  sum  of  the  legs  and  the 
hypotenuse. 

1116.  All  chords  of  a  circle  which  touch  an  interior  concentric 
circle  are  equal,  and  are  bisected  at  the  points  of  contact. 

Theorem  50.  In  equal  circles  central  angles  have  the  same 
ratio  as  their  intercepted  arcs. 

COT.  1.    A  central  angle  is  measured  by  its  intercepted  arc. 

Theorem  51.    Parallels  intercept  equal  arcs  on  a  circle. 

Theorem  52.  An  inscribed  angle,  or  one  formed  by  a  tangent 
and  a  chord  is  measured  by  one-half  its  intercepted  arc. 

Theorem  52a.  The  mid-point  of  the  hypotenuse  of  a  right 
triangle  is  equidistant  from  the  three  vertices. 

What  point  in  the  circumscribed  circle  is  the  mid-point  of  the  hypotenuse? 

Theorem  53.  An  angle  whose  vertex  is  inside  the  circle  is  mea- 
sured by  half  the  sum  of  the  arcs  intercepted  by  it  and  its  vertical. 

Theorem  54.  An  angle  whose  vertex  is  outside  the  circle  is 
measured  by  half  the  difference  of  its  intercepted  arcs. 

Theorem  54a.  The  opposite  angles  of  a  quadrilateral  inscribed 
in  a  circle  are  supplementary. 

(Proof  left  to  the  pupil.) 


THE  CIRCLE  279 

Cor.  1.    A  quadrilateral  is  inscriptible  if  its  opposite  angles  are 
supplementary. 

Suggestion:   Show,  by  the  method  of  exclusion,  that  the  fourth  vertex  of  the 
quadrilateral  lies  on  the  circle  passing  through  three  of  its  vertices. 

EXERCISES.    SET  LXXXVI.     MEASUREMENT  OF  ANGLES 

Numeric 

1117.  Find  the  value  of  an  angle  which  (a)  is  inscribed,  and  in- 
tercepfcs  an  arc  of  160°,  (6)  is  inscribed  in  a  segment  of  250°. 

1118.  If  the  tangents  from  a  point  to  a  circle  make  an  angle  of 
60°,  what  are  the  values  of  the  arcs  they  intercept?    What  if  the 
angle  is  a  right  angle? 

1119.  Find  the  angle  whose  sides  are  tangents  drawn  from  a 
point  whose  distance  from  the  center  of  the  circle  is  the  diameter 
of  that  circle 

1120.  An  angle  between  two  chords  intersecting  inside  a  circle 
is  35°,  its  intercepted  arc  is  25°  18';  find  the  arc  intercepted  by  its 
vertical. 

1121.  A  triangle  is  inscribed  in  a  circle,  and  another  triangle  is 
circumscribed  by  drawing  tangents  at  the  vertices  of  the  inscribed 
triangle.    The  angles  of  the  inscribed  triangle  are  40°,  60°,  and  80°. 
Find  all  the  other  angles  of  the  figure. 

1122.  The  arcs  subtended  by  three  consecutive  sides  of  a  quad- 
rilateral are  87°,  95°,  115°;  find  the  angles  of  the  quadrilateral; 
the  angles  made  by  the  intersection  of  the  diagonals,  and  the 
angles  made  by  the  opposite  sides  of  the  quadrilateral  when 
produced. 

1123.  Three  consecutive  angles  of  an  inscribed  quadrilateral 
are  140°  30',  80°  30',  and  39°  30'.    Find  the  numbers  of  degrees  in 
the  arcs  subtended  by  the  four  sides  if  the  arc  intercepted  by  the 
largest  angle  is  divided  into  parts  in  the  ratio  of  4  to  5. 

1124.  Three  consecutive  angles  of  a  circumscribed  quadrilateral 
are  85°,  122°,  111°.  Find  the  number  of  degrees  in  each  angle  of  the 
inscribed  quadrilateral  made  by  joining  the  points  of  contact  of  the 
sides  of  the  circumscribed  quadrilateral. 

1125.  The  points  of  tangency  of  a  quadrilateral,  circumscribed 


280  PLANE  GEOMETRY 

about  a  circle,  divide  the  circumference  into  arcs,  which  are  to 
each  other  as  4,  6,  10,  and  16.  Find  the  angles  of  the  quadrilateral. 
1126.  If  the  sides  AB  and  BC  of  an  inscribed  quadrilateral 
A  BCD  subtend  arcs  of  60°  and  130°,  respectively,  and  the  diagonals 
form  $.AED  =  70°,  find  the  number  of  degrees  in  (a)  15,  (b) 
(c)  each  angle  of  the  quadrilateral. 

0. ^  1127.  In  this  figure  ££= 41°, 

=  65°,  and  $BCD  =  97°.  Find  the 
number  of  degrees  in  each  of  the  other 
angles,  and  determine  whether  or  not 
CD  is  a  diameter. 

1128.  In  this  figure  <£ra  =  62°  and 
'B  <£n  =  28°.  Find  the  number  of  de- 
grees in  each  of  the  other  angles,  and  deter- 
mine whether  or  not  AB  is  a  diameter. 

1129.  At  the   vertices   of     an   inscribed 
quadrilateral  tangents  are  drawn  to  the  cir- 
cle, forming  a  circumscribed   quadrilateral. 

The  arcs  subtended  by  the  sides  of  the  in-  T 

scribed  quadrilateral  are  in  the  ratio  of  3  to  4  to  5  to  8. 

(a)  Find  the  angles  of  each  quadrilateral. 

(b)  Find  the  angles  between  the  diagonals  of  the  inscribed 
quadrilateral. 

(c)  Find  the  angles  between  the  opposite  sides  of  the  inscribed 
quadrilateral  produced  to  intersect. 

(d)  Find  the  angles  between  the  sides  of  the  inscribed  and  those 
of  the  circumscribed  quadrilateral. 

1130.  The  vertices  of  a  quadrilateral  inscribed  in  a  circle  divide 
the  circumference  into  arcs  which  are  to  each  other  as  1,  2,  3,  and  4. 
Find  the  angles  between  the  opposite  sides  of  the  quadrilateral. 

1131.  The  sides  of  an  inscribed  quadrilateral  subtend  arcs  in  the 
ratio  (a)  1  to  2  to  3  to  4,  (6)  3  to  5  to  7  to  9.    How  many  degrees  in 
each  angle  of  the  quadrilaterals  in  (a)  and  (6)? 

1132.  The  bases  of  an  inscribed  isosceles  trapezoid  subtend  arcs 
of  100°  and  120°.    How  many  degrees  in  each  angle  of  the  trapezoid 
(a)  if  the  bases  are  on  the  same  side  of  the  center,  (6)  if  they  are 
on  opposite  sides  of  the  center? 


THE  CIRCLE  281 


Theoretic 

1133.  The  angle  formed  by  two  tangents  is  equal  to  twice  the 
angle  between  the  chord  of  contact  *  and  the  radius  drawn  to  a 
point  of  contact. 

1134.  If  the  tangents  drawn  from  an  exterior  point  to  a  circle 
form  an  angle  of  120°,  the  distance  of  the  point  from  the  center  is 
equal  to  the  sum  of  the  tangents. 

1135.  An  isosceles  trapezoid  is  inscriptible;  that  is,  a  circle  can 
be  circumscribed  about  it. 

1136.  If  in  a  circle  two  chords  are  drawn,  and  the  mid-point  of 
the  arc  subtended  by  one  chord  is  joined  to  the  extremities  of  the 
other  chord,  the  two  triangles  thus  formed  are  mutually  equi- 
angular, and  the  quadrilateral  thus  formed  is  inscriptible. 

1137.  If  A,  B,  C,  Ai,  BI,  Ci  are  six  points  in  a  circumference, 
such  that  AB  is  parallel  to  A\Bi  and  AC  is  parallel  to  AiCi,  then 
BCi  is  parallel  to  B£. 

1138.  Let  A  be  any  point  of  a  diameter,  B  the  extremity  of  a 
radius  perpendicular  to  the  diameter,  P  the  point  in  which  BA 
meets  the  circumference,  C  the  point  in  which  the  tangent  through 
P  meets  the  diameter  produced.    Prove  that  AC  =  PC. 

1139.  If  two  circles  touch  internally,  and  the  diameter  of  the 
smaller  is  equal  to  the  radius  of  the  larger,  the  circumference  of  the 
smaller  bisects  every  chord  of  the  larger  which  can  be  drawn 
through  the  point  of  contact. 

dll40.  Two  circles  touch  internally  in  the  point  P,  and  A  B  is  a 
chord  of  the  larger  circle  touching  the  smaller  in  the  point  C. 
Prove  that  PC  bisects  the  angle  APE. 

1141.  If  two  circles  intersect  at  the  points  A  and  B}  and  through 
A  a  variable  secant  be  drawn  cutting  the  circles  in  C  and  D,  the 
angle  CBD  is  constant  for  all  positions  of  the  secant. 

1142.  If  two  circles  are  tangent  externally,  the  corresponding 
sects  of  two  lines  drawn  through  the  point  of  contact  and  ter- 
minated by  the  circles  are  proportional. 

6    By  the  chord  of  contact  is  meant  the  sect  joining  the  points  of  contact 
of  a  pair  of  tangents. 


282  PLANE  GEOMETRY 

1143.  If  two  circles  are  tangent  to  each  other  and  a  sect  be 
drawn  through  the  point  of  tangency,  terminating  in  the  circles, 
the  diameters  from  the  extremities  of  this  sect  are  parallel. 

Case  I.    Circles  tangent  externally. 
Case  II.    Circles  tangent  internally. 

1144.  If  two  circles  are  tangent  to  each  other  and  a  sect  be 
drawn  through  the  point  of  tangency  and  terminating  in  the 
circles,  tangents  at  the  extremities  of  this  sect  are  parallel. 

Case  I,  the  circles  tangent  externally;  and  Case  II,  tangent 
internally. 

dl!45.  If  two  sects  OA  and  OB,  not  in  a  straight  line,  are  divided 
in  C  and  D,  respectively,  so  that  03-OC=05-OD,  then  A,  B,  C,  D 
are  concyclic,  that  is,  lie  on  the  same  circle. 

dl!46.  The  altitudes  of  a  triangle  bisect  the  angles  of  the  tri- 
angle determined  by  their  feet,  i.e.,  the  angles  of  the  pedal  triangle. 

dl!47.  The  feet  of  the  medians  and  the  feet  of  the  altitudes  of  a 
triangle  are  concyclic. 

Hint:  Pass  a  circle  through  the  feet  of  the  medians,  then  prove  that  the 
foot  of  any  one  of  the  three  altitudes  will  lie  on  this  circle. 

1148.  If  two  circles  are  tangent  externally,  and  a  secant  is  drawn 
through  the  point  of  contact,  the  chords  formed  are  proportional 
to  the  radii. 

1149.  If  C  is  the  mid-point  of  AB,  and  chord  CD  cuts  chord  AB 
.         W  =  CA 

dl!50.  If  two  circles  are  tangent  externally,  the  common  tangent 

is  a  mean  proportional  between  the  diameters. 

(11151.  The 
line  joining  the 
extremities  of 
two  parallel  radii 

D\ ^ J/L    V"   1,     ^^>P  of  two  circles 

passes  through 
the  direct  center 
of  similitude  if 
the  radii  have  the 
same  direction, 

and  through  the  inverse  center  if  the  radii  have  opposite  directions. 


THE  CIRCLE  283 

dl!52.  Taking  any  point  as  a  center  of  similitude  of  two  circles, 
the  two  radii  of  one  of  them,  drawn  to  its  points  of  intersection 
with  any  other  line  passing  through  that  center  of  similitude,  are 
parallel,  respectively,  to  the  two  radii  of  the  other,  drawn  to  its 
intersections  with  the  same  line. 

Hint:   Use  an  indirect  proof  depending  upon  Ex.  1151. 

dl!53.  All  secants,  drawn  through  a  direct  center  of  similitude 
P  of  two  circles,  cut  the  circles  in  points  whose  distances  from  P, 
taken  in  order,  form  a  proportion. 

dl!54.  If  in  the  last  exercise,  the  line  of  centers  cuts  the  circles 
in  points  A,  B,  C,  D,  and  any  other_secant  through  P  cuts  the 
circle  in  points  M,  N,  R,  S,  prove  that  PN»PR  is  constant  and  equal 
to  PB-PC. 

dl!55.  The  common  external  tangents  to  two  circles  pass 
through  the  direct  center  of  similitude,  and  the  common  interior 
tangents  pass  through  the  inverse  center  of  similitude. 

What  method  of  drawing  the  common  tangents  to  two  circles 
may  be  derived  from  this  fact? 

1156.  ABC  is  an  isosceles  triangle  inscribed  in  a  circle,  BD  a 
chord  drawn  from  its  vertex  cutting  the  base  in  any  point  E. 

_.         'BD    AB 

Prove  === 

AB    BE 

1157.  If  two  circles  are  tangent  internally,  all  chords  of  the 
greater  circle  drawn  from  the  point  of  contact  are  divided  pro- 
portionally by  the  circumference  of  the  smaller. 

1158.  If  two  circles  touch  at  M,  and  through  M  three  lines  are 
drawn  meeting  one  circle  in  A,  B,  C,  and  the  other  in  D,  E,  F, 
respectively,  the  triangles  ABC  and  DEF  are  similar. 

Locus 

1159.  An  angle  of  60°  moves  so  that  both  of  its  sides  touch  a 
fixed  circle  of  radius  5  ft.    What  is  the  locus  of  the  vertex? 

1160.  Find  the  locus  of  the  mid-point  of  a  chord  drawn  through 
a  given  point  within  a  given  circle. 

1161.  Through  a  point  A  on  a  circle  chords  are  drawn.    On  each 
one  of  these  chords  a  point  is  taken  one-third  the  distance  from  A 
to  the  end  of  the  chord.    Find  the  locus  of  these  points. 


284  PLANE  GEOMETRY 

1162.  The  locus  of  the  vertex  of  a  triangle,  having  a  given  base 
and  a  given  angle  at  the  vertex,  is  the  arc  which  forms,  with  the 
base,  a  segment  capable  of  containing  the  given  angle. 

1163.  Find  the  locus  of  the  points  of  contact  of  tangents  drawn 
from  a  given  point  to  a  given  set  of  concentric  circles. 

1164.  A  variable   chord   passes,   when  prolonged,   through  a 
fixed  point  outside  a  given  circle.    What  is  the  locus  of  the  mid- 
point of  the  chord? 

1166.  Upon  a  sect  AB  a  segment  of  a  circle  containing  240°  is 
constructed,  and  in  the  segment  any  chord  CD  subtending  an  arc 
of  60°  is  drawn.  Find  the  locus  of  the  intersection  of  AC  and  BD, 
and  also  of  the  intersection  of  AD  and  BC. 

1166.  The  locus  of  the  centers  of  circles  inscribed  in  triangles 
having  a  given  base  and  a  given  angle  at  the  vertex  is  the  arc 
which  forms  with  the  base  a  segment  capable  of  containing  a  right 
angle  plus  half  the  given  angle  at  the  vertex. 

1167.  The  locus  of  the  intersections  of  the  altitudes  of  triangles 
having  a  given  base  and  a  given  angle  at  the  vertex  is  the  arc 
forming  with  the  base  a  segment  capable  of  containing  an  angle 
equal  to  the  supplement  of  the  given  angle  at  the  vertex. 

dl!68.  Find  the  locus  of  a  point  from  which  two  circles  subtend* 
the  same  angle. 

1169.  If  A  and  B  are  two  fixed  points  on  a  given  circle,  and  P 
and  Q  are  the  extremities  of  a  variable  diameter  of  the  same 
circle,  find  the  locus  of  the  point  of  intersection  of  the  lines 
AP  and  BQ. 

dllTO.  The  lines  li  and  k  meet  at  right  angles  in  a  point  A .  0  is 
any  fixed  point  on  k>  Through  0  draw  a  line  meeting  li  in  B. 
P  is  a  varying  point  on  this  line  such  that  OB'OP  is  constant. 
Plot  the  locus  of  P  as  the  line  swings  about  0  as  a  pivot. 

Theorem  66.  A  tangent  is  the  mean  proportional  between  any 
secant  and  its  external  sect,  when  drawn  from  the  same  points 
to  a  circle. 

*  If  tangents  from  the  same  point  to  two  circles  form  equal  angles,  the  circles 
are  said  to  subtend  equal  angles  from  that  point. 


THE  CIRCLE 


285 


Cor.  1.  The  product  of  a  sec- 
ant and  its  external  sect 
from  a  fixed  point  out- 
side a  circle  is  constant. 

Hint:   Draw  tangent  PIT. 
What  is  the  constant  in  this  corollary? 
State  the  corollary  in  another  way. 

Theorem  55a.    If  chords  intersect  inside  a  circle,  the  product 
of  their  sects  is  constant, 

Prove  by  means  of  similar  triangles. 

Applying  the  principle  of  continuity,  Theorem  55,  Cor.  1,  and 
Theorem  55a  can  be  stated  as  a  single  theorem.    State  them  so. 

Theorem  556.  The  square  of  the  bi- 
sector of  an  angle  of  a  triangle  is  equal 
to  the  product  of  the  sides  of  this  angle, 
diminished  by  the  product  of  the  sects 
made  by  that  bisector  on  the  third  side. 

Given:    AABC  with  bisector  fo  cutting  AC  at  D 

into  sects  q  and  r. 
Prove:   tbz  =  ac—  qr. 

PROOF 

Circumscribe  QO  about  ABC,  and 
extend  BD  to  E  in  00  and  draw 
chord  CE. 

LetDE^s. 

(1)  &BDA  c/3  ABCE  (1)  Why? 

^-toTi-l  (2>  Why? 

(3)  .-.ac=fe2+fcs  (3)  Why? 

(4)  But  tbs=qr  (4)  Why? 

(5)  /.  ac  =  tb*+qr  (5)  Why? 

(6)  .-.  tb*=ac-qr  (6)  Why? 

Theorem  55c.  In  any  triangle  the  product 
of  two  sides  is  equal  to  the  product  of  the 
diameter  of  the  circumscribed  circle  and  the 
altitude  on  the  third  side. 
Hint:   Prove  AABX ~  AEBC. 

Consider  the  special  case  where  3-  B  is  a  right 
angle  and  evolve  a  formula  for  hb. 


286  PLANE  GEOMETRY 

Cor.  1.  If  R  denote  the  radius  of  the  circle  circumscribed  about 
a  triangle  whose  sides  are  a,  b,  c,  and  semiperimeter  s, 
then 

R= abc 

4\A(s  — a)(s  —  b)(s  — c)* 
In  the  figure  for  Theorem  55c,  ac=hbD. 


2     

But  hb=7\fs(s  — a)(a  —  6)(s  — c) 

ac  a&c 


s  -a)(s  -6)(s  -c)     4VsO  -a)(s  -b)(s  -c) 
b 

EXERCISES.    SET  LXXXVII      METRIC  RELATIONS 

Numeric 

1171.  A  point  P  is  10  in.  from  the  center  of  a  circle  whose  radius 
is  6  in.    Find  the  length  of  the  tangent  from  P  to  the  circle. 

1172.  The  length  of  a  tangent  from  P  to  a  circle  is  7  in.,  and  the 
external  sect  of  a  secant  is  4  in.    Find  the  length  of  the  whole 
secant. 

1173.  A  point  P  is  8  in.  from  the  center  of  a  circle  whose  radius 
is  4.    Any  secant  is  drawn  from  P,  cutting  the  circle.    Find  the 
product  of  the  whole  secant  and  its  external  sect. 

1174.  From  the  same  point  outside  a  circle  two  secants  are 
drawn.    If  one  secant  and  its  external  sect  are  24  and  15,  respec- 
tively, and  the  external  sect  of  the  other  is  7,  find  that  secant. 

1175.  Two  chords  intersect  within  a  circle.    The  sects  of  one 
are  m  and  n  and  one  sect  of  the  other  is  p.   Find  the  remaining  sect. 

1176.  If  a  tangent  and  a  secant  drawn  from  the  same  point  to  a 
circle  measure  6  in.  and  18  in.,  respectively,  how  long  is  the  ex- 
ternal sect  of  the  secant? 

1177.  Two  secants  are  drawn  from  a  common  point  to  a  circle. 
If  their  external  sects  are  12  and  9,  and  the  internal  sect  of  the 
first  is  8,  what  is  the  length  of  the  second? 

1178.  The  radius  of  a  circle  is  13  in.     Through  a  point  5  in. 
from  the  center  a  chord  is  drawn.    What  is  the  product  of  the  two 


THE  CIRCLE  287 

sects  of  the  chord?    What  is  the  length  of  the  shortest  chord  that 
can  be  drawn  through  that  point? 

1179.  One  sect  of  a  chord  through  a  point  3.5  units  from  the 
center  of  a  circle  is  2  units  in  length.    If  the  diameter  of  the  circle 
is  12  units,  what  is  the  length  of  the  other  sect  of  the  chord? 

1180.  The  radius  of  a  circle  is  2  units.    If  through  a  point  P, 
4  units  from  the  center,  secant  PQR  is  drawn,  and  QR  is  one  unit, 
what  is  the  length  of  PQ? 

1181.  A  ABC  is  inscribed  in  a  circle  of  radius  5  in.    Find  the 
altitude  to  BC  if  AB  is  4,  and  AC  is  5  in. 

1182.  The  sides  of  a  triangle  are  4,  13,  and  15,  respectively. 
Find  the  radius  of  the  circumscribed  circle. 

1183.  In  Aa6c,  a  =  20,  6  =  15,  and  the  projection  of  b  upon  c 
(pbc)  is  9.    Find  the  radius  of  the  circumscribed  circle. 

1184.  In  Aa&c,  a  =  9  and  6=12.    Find  c  if  the  diameter  of  the 
circumscribed  circle  is  15. 

1185.  The  sides  of  a  triangle  are  18,  9,  and  21,  respectively. 
Find  the  angle  bisector  corresponding  to  21. 

1186.  The  sides  of  a  triangle  are  21,  14,  and  25,  respectively. 
Find  the  angle  bisector  corresponding  to  25. 

1187.  The  sides  of  a  triangle  are  22,  11,  and  21,  respectively. 
Find  the  angle  bisector  corresponding  to  21. 

1188.  The  sides  of  a  triangle  are  6,  3,  and  7,  respectively.    Find 
the  angle  bisector  corresponding  to  7. 

1189.  In  a  triangle  the  sides  of  which  are  48,  36,  and  50,  where 
do  the  bisectors  of  the  angles  intersect  the  sides?    What  are  the 
lengths  of  the  angle  bisectors? 

1190.  In  each  of  the  Exercises  1181  to  1189  what  kind  of  triangle 
is  involved? 

Construction 

1191.  Construct  the  mean  proportional  between  two  given  sects, 
using  in  turn  the  methods  suggested  by  the  following  propositions : 

(a)  39  Cor.  1,  (6)  39  Cor.  4,  (c)  55. 

1192.*  Construct  a  square  equal  in  area  to  that  of  a  given:    (a) 
rectangle;  (6)  triangle;  (c)  trapezoid. 

*  For  discussion  and  illustrations  of  the  type  of  analysis  applicable  see 
pp.  318  to  321,  and  Problems  17,  19,  21,  22,  26,  27,  Chapter  VII. 


288  PLANE  GEOMETRY 

1193.  Draw  through  a  given  external  point  P  a  secant  PAR  to  a 
given  circle  so  that  AB2  =  PA-PB.  VII. 

dl!94.  Draw  through  one  of  the  points  of  intersection  of  two 
given  intersecting  circles  a  common  secant  of  given  length.  VII. 

dl!95.  From  a  point  outside  a  circle  draw  a  secant  whose  exter- 
nal sect  is  equal  to  one-half  the  secant. 

Locus 

dl!96.  Given  the  fixed  base  of  a  triangle  and  the  sum  of  the 
squares  of  the  other  two  sides,  describe  the  locus  of  the  vertex. 

dl!97.  Repeat  Ex.  1 196,  given  the  difference  of  the  squares  of  the 
other  two  sides. 

dl!98.  Through  P  any  PMN  is  drawn,  cutting  a  circle  K  in  M 
and  N,  and  P  moves  so  that  the  product  of  the  sects  PM  PN  has 
the  constant  value  k2.  Find  the  locus  of  P. 

Theoretic 

1199.  If  chord  AB  bisects  chord  CD,  either  sect  of  chord  CD 
is  a  mean  proportional  between  the  sects  of  AB. 

1200.  If  two   circles  intersect,  their  common  chord  produced 
bisects  the  common  tangents. 

1201.  In  the  diameter  of  a  circle  points  A  and  B  are  taken 
equally  distant  from  the  center,  and  joined  to  a  point  P  in  the 
circumference.     Prove  that  ^AP2+BP2  is  constant  for  all  posi- 
tions of  P. 

1202.  If  a  tangent  is  limited  by  two  other  parallel  tangents  to 
the  same  circle,  the  radius  of  the  circle  is  the  mean  proportional 
between  its  sects. 

1203.  The  tangents  to  two  intersecting  circles,  drawn  from  any 
point  in  their  common  chord  produced,  are  equal. 

d!204.  The  sum  of  the  squares  of  the  diagonals  of  a  trapezoid  is 
equal  to  the  sum  of  the  squares  of  the  legs  plus  twice  the  product 
of  the  bases. 

1205.  In  an  inscribed  quadrilateral  the  product  of  the  diagonals 
is  equal  to  the  sum  of  the  products  of  the  opposite  sides.  (Ptolemy's 
Theorem.) 

d!206.  If  the  opposite  sides  of  an  inscribed  hexagon  intersect, 
they  determine  three  collinear  points.  ("  Mystic  Hexagram," 
discovered  by  Pascal  when  he  was  16  years  of  age.) 


THE  CIRCLE  289 

d!207.  If  a  circumference  intersects  tha  sides  a,  b,  c,  of  a  AABC 
in  the  points  Ai  and  A2,  #1  and  B2,  Ci  and  C2,  respectively,  then 
Ad  BAi  CBi  ACZ  BAz  CB2  . , 

L    (Camot 


Theorem  56.  A  circle  may  be  circumscribed  about,  and  in- 
scribed within,  any  regular  polygon 

Cor.  1.    An  equilateral  polygon  inscribed  in  a  circle  is  regular. 
Cor.  2.    An    equiangular  polygon   circum- 
scribed about  a  circle  is  regular. 

Cor.  3.    The  area  of  a  regular  polygon  is  A> 
equal  to  half  the  product  of  its  apo- 
them  and  perimeter. 
Suggestion:    What  is  the  area  of  AAOBf  ff 

Theorem  57.  If  a  circle  is  divided  into  any  number  of  equal 
arcs,  the  chords  joining  the  successive  points  of  division  form  a 
regular  inscribed  polygon;  and  the  tangents  drawn  at  the  points 
of  division  form  a  regular  circumscribed  polygon. 

Cor.  1.  Tangents  to  a  circle  at  the  vertices  of  a  regular  inscribed 
polygon  form  a  regular  circumscribed  polygon  of  the 
same  number  of  sides. 

Cor.  2.  Lines  drawn  from  each  vertex  of  a  regular  inscribed 
polygon  to  the  mid-points  of  the  adjacent  arcs  sub- 
tended by  its  sides  form  a  regular  inscribed  polygon 
of  double  the  number  of  sides. 

Cor.  3.  Tangents  at  the  mid-points  of  the  arcs  between  con- 
secutive points  of  contact  of  the  sides  of  a  regular 
circumscribed  polygon,  form  a  regular  circumscribed 
polygon  of  double  the  number  of  sides. 

Cor.  4.  The  perimeter  of  a  regular  inscribed  polygon  is  less 
than  that  of  a  regular  inscribed  polygon  of  double 
the  number  of  sides;  and  the  perimeter  of  a  regular 
circumscribed  polygon  is  greater  than  that  of  a 
regular  circumscribed  polygon  of  double  the  number 
of  sides. 
19 


290 


PLANE  GEOMETRY 


Cor.  5.  Tangents  to  a  circle  at  the  mid-points  of  the  arcs  sub- 
tended by  the  sides  of  a  regular  inscribed  polygon, 
form  a  regular  circumscribed  polygon,  of  which  the 
sides  are  parallel  to  those  of  the  original  polygon  and 
the  vertices  lie  on  the  prolongations  of  the  radii  of  the 
one  inscribed. 

Suggestions:  Show  that  AB  and  AiBi  are 
both  perpendicular  to  OP  and  are, 
therefore,  parallel. 
Show   .*.   that    %.A  =  Xt.A1,  4#  = 

4#i,  .... 

What  kind  of  n-gon  is  then  circum- 
scribed? 

BP=BT  and  £iPi=£i7\  and, 
therefore,  B  and  BI  lie  on  bisector 
of  X,POT. 

Radius  OBi  bisects  4. POT. 

Theorem  58.  A  regular  polygon  the  number  of  whose  sides 
is  3*2"  may  be  inscribed  in  a  circle. 

Theorem  59.  If  in  represent  the  side  of  a  regular  inscribed 
polygon  of  n  sides  and  i2n  the  side  of  one  of  2n  sides  and  r  the 

radius  of  the  circle,  i2n=\2r2  -rV/4r2  -in2. 

Theorem  60.  //  in  represent  the  side  of  a  regular  inscribed 
polygon  of  n  sides,  cn  that  of  a  regular  circumscribed  polygon  of  n 


2ri, 


sides,  and  r  the  radius  of  the  circle,  cn= 


Theorem  61.  The  perimeters  of  regular  polygons  of  the  same 
number  of  sides  compare  as  their  radii  and  also  as  their  apothems. 

Theorem  62.  Circumferences  have  the  same  ratio  as  their 
radii. 

Cor.  1.  The  ratio  of  any  circumference  to  its  diameter  is 
constant. 

Cor.  2.    In  any  circle  c=Zirr. 

Theorem  63.    The  value  of  TT  is  approximately  3.14159. 

Theorem  64.  The  area  of  a  circle  is  equal  to  one-half  the 
product  of  its  radius  and  its  circumference. 

Cor.  1.  The  area  of  a  circle  is  equal  to  TT  times  the  square  of 
its  radius. 


THE  CIRCLE  291 

Cor.  2.    The  areas  of  circles  compare  as  the  squares  of  their 

radii. 
Cor.  3.    The  area  of  a  sector  is  equal  to  half  the  product  of  its 

radius  and  its  arc. 

(Proof  left  to  the  student.) 

NOTE.  —  Cor.  3  does  not  suggest  the  most  convenient  method  of  determining 
the  area  of  a  sector.    Suggest  a  more  convenient  one. 

A  segment  of  a  circle  is  a  portion  of  it  bounded  by  an  arc  and  its 
subtending  chord.  Similar  sectors  and  similar  segments  are  those 
of  which  the  arcs  contain  the  same  number  of  degrees. 

Cor.  4.  Similar  sectors  and  similar  segments  compare  as  the 
squares  of  their  radii. 

Suggestions:  How  do  circles 
0  and  Oi  compare  in  area? 
How,  then,  would  like  parts 
of  them  compare? 

Why  are  similar  sectors 
like  parts? 

Why  is  AAOB™ 
AAtOiBJ  How,  therefore, 
do  the  triangles  compare 
in  area? 

Justify  the  folio  wing  alge- 

braic statements  which  form  the  basis  of  proof  here: 
sec.  AOB         R2\      &AOB       .   sec.AOB 


.  AOB   _(R2\_  &AOB       .   sec.AOB 
^iOiBi    \fV~AArfS5      '    AAOB 
seg.AOB_seg.AiOiBi        seg.  AOB  _  &AOB  ^R* 
'    AAOB  =  AAiOitfi    °r  seg.  A&Bi  "  A4i0i#i  ~~  r2' 

EXERCISES.      SET  LXXXVIII.     MENSURATION  OF  THE  CIRCLE 

Numeric 


1208.  Find 
follows  : 
(a)  9  in. 
(6)   12  in. 
1209.  Find 
follows  : 
(a)  15 

(6)     7T2 

the 

(c) 
(<*) 

the 

(c) 
(d) 

circumferences   of   circles 

5.9  in.           (e)  2|  ft. 
7.3  in.           (/)  3|-  in. 
diameters  of   circles  with 

2wr               (e)  188.496  in. 
?7ra2             (/)  219.912  in. 

with   diameters  as 

(g)  29  centimeters 
(h)  47  millimeters 
circumferences  as 

(g)  3361.512  in. 
(h)  3173.016  in. 

292  PLANE  GEOMETRY 

1210.  Find  the  diameter  of  a  carriage  wheel  that  makes  264 
revolutions  in  going  half  a  mile. 

1211.  The  diameter  of  a  bicycle  wheel  is  28  in.     How  many 
revolutions  does  the  wheel  make  in  going  10  mi.? 

1212.  Find  the  radii  of  circles  with  circumferences  as  follows: 
(a)  77r  (c)   15.708  in.     (e)  18.8496  in.   (g)  345.576  ft. 
(6)  3^-TT          (d)  21.9912  in.    (/)  125.664  in.  (h)  3487.176  in. 

1213.  Find  the  radius  of  a  circle  whose  circumference  is  m  units. 

1214.  An  arc  of  a  certain  circle  is  100  ft.  long  and  subtends  an 
angle  of  25°  at  the  center.    Compute  the  radius  of  the  circle  correct 
to  one  decimal  place. 

1215.  The  circumference  of  a  circle  is  10.    Find  the  circumfer- 
ence of  one  having  twice  the  area  of  the  original. 

1216.  Find  the  central  angle  of  a  sector  whose  perimeter  is  equal 
to  the  circumference. 

1217.  Find  the  areas  of  circles  with  diameters  as  follows: 
(a)  16a6         (c)   2.5ft.  (e)  3f  yd.          (g)  3  ft.  2  in. 
(6)  24vr2          (d)  7.3  in.          (/)  4f  yd.          (h)  4  ft.  1  in. 

1218.  Find  the  area  of  circles  with  radii  as  follows: 

(a)  5x  (c)   27  ft.  (e)  3£  in.  (g)  2  ft.  6  in. 

(b)  27r  (d)  4.8  ft.  (/)  4|  in.  (h)  7  ft.  9  in. 

1219.  Find  the  radii  of  circles  with  areas  as  follows : 

(a)  7ra264         (c)ir  (e)  12.5664         (g)  78.54 

(b)  47rmV5     (d)  27T  (/)  28.2744        (h)  113.0976 

1220.  Find  the  areas  of  circles  with  circumferences  as  follows : 

(a)  27r  (c)   ira  (e)  18.8496  in.    (g)  333.0096  in. 

(b)  47r  (d)  147ra2  (/)  329.868  in.   (h)  364.4256  in. 

1221.  Find  the  area  of  a  circle  whose  circumference  is  C. 

1222.  Find  the  area  of  a  sector  whose  radius  is  5  and  whose 
central  angle  is  40°. 

1223.  Find  the  area  of  a  fan  that  opens  out  into  a  sector  of  120°, 
the  radius  of  which  is  9^  in. 

1224.  The  arc  of  a  sector  of  a  circle  2j  in.  in  diameter  is  If  in, 
What  is  the  area  of  the  sector? 

1225.  Find  the  central  angle  of  a  sector  whose  area  is  equal  to 
the  square  of  the  radius. 


THE  CIRCLE 


293 


1226.  Find  the  circumference  of  a  circle  whose  area  is  S. 

1227.  A  circle  has  an  area  of  60  sq.  in.    Find  the  length  of  an 
arc  of  40°. 

1228.  Find  the  radius  of  a  circle  equivalent  to  a  square  the  side 
of  which  is  6. 

1229.  The  circumferences  of  two  concentric  circles  are  30  and  40, 
respectively.    Find  the  area  bounded  by  the  two  circumferences 
by  the  shortest  method  you  know. 

1230.  In  an  iron  washer  here  shown,  the  dia- 
meter of  the  hole  is  1|-  in.,  and  the  width  of 
the  washer  is  -f  in.    Find  the  area  of  one  face 
of  the  washer. 

1231.  The  area  of  a  fan  which  opens  out 
into  a  sector  of  111°  is  96.866  sq.  in.    What  is 
the  radius?    (Use  ir  =  3.1416.    Why?) 

1232.  The  radius  of  a  circle  is  10  ft.     Two  parallel  chords  are 
drawn,  each  equal  to  the  radius.    Find  that  part  of  the  area  of  the 
circle  lying  between  the  parallel  chords. 

1233.  A  square  is  inscribed  in  a  circle  of  radius  10.    Find  the 
area  of  the  segment  cut  off  by  a  side  of  the  square. 

1234.  Find    a    semicircle    equivalent    to  an 
equilateral  triangle  whose  side  is  5. 

1235.  A  kite  is  made  as  shown  in  the  dia- 
gram, the  semicircle  having  a  radius  of  9  in., 
and  the  triangle  a  height  of  25  in.    Find  the 
area  of  the  kite. 

1236.  Two  circles  are  tangent  internally,  the 
ratio  of  their  radii  being  2  to  3.    Compare  their 
areas,  and  also  the  area  left  in  the  larger  circle 
with  that  of  each  of  the  circles. 

1237.  A   reservoir  constructed  for  irrigation 
purposes  sends  out  a  stream  of  water  through 

a  pipe  3  ft.  in  diameter.  The  pipe  is  1000  ft.  long.  How  many 
times  must  it  be  filled  if  it  is  to  discharge  10,000  acre-feet  of 
water?  (An  acre-foot  of  water  is  the  amount  required  to  cover 
1  acre  to  a  depth  of  1  ft.) 


294  PLANE  GEOMETRY 

1238.  Each  side  of  a  triangle  is  2n  centimeters,  and  about  each 
vertex  as  center,  with  radius  of  n  centimeters,  a  circle  is  described. 
Find  the  area  bounded  by  the  three  arcs  that  lie  outside  the  triangle, 
and  the  area  bounded  by  the  three  arcs  that  lie  inside  the  triangle. 

1239.  From  a  point  outside  a  circle  whose  radius  is  10,  two 
tangents  are  drawn.     Find  the  area  bounded  by  the  tangents 
and  the  circumference,  if  they  include  an  angle  of  120°.     Find 
both  results. 

1240.  Upon  each  side  of  a  square  as  a  diameter  semicircles  are 
described  inside  the  square.    If  a  side  of  the  square  is  s,  find  the 
sum  of  the  areas  of  the  four  leaves. 

A      1241.  Find  the  area  bounded  by  three  arcs  each 
of  60°  and  radius  5  if  the  convex  sides  of  the  arcs  are 
turned  toward  the  area. 
1242.  Find  the  area  bounded  by  three 
arcs  each  of  60°  and  radius  5  if  the 
concave  sides  of  the  arcs  are  turned  toward  the 
area. 

dt!243.  The  flywheel  of  an  engine  is  connected 
by  a  belt  with  a  smaller  wheel  driving  the  machinery  of  a  mill.  The 
radius  of  the  flywheel  is  7  ft.,  and  of  the  driving  wheel  is  21  in.  (a) 
How  many  revolutions  does  the  smaller  wheel  make  to  one  of  the 
larger  wheel?  (6)  The  distance  between  the  centers  is  10  ft.  6  in. 
What  is  the  length  of  the  belt  connecting  the  two  wheels  if  it  is 
not  crossed?  (c)  If  it  is  crossed? 

1244.  Given  a  circle  whose  radius  is  16,  find  the  perimeter  and 
the  area  of  the  regular  inscribed  octagon. 

1245.    The   following    is    Ceradini's    approximate 
method  of  constructing  a  sect  equal  in  length  to  a 
circle:  Draw  diameter  AB  and  tan- 
gent BK  at  B.     Draw  OC  making 
-K     <£CO£  =  30°.  Make  CD  =30£.  Draw 
AD,  and  prolong  it,  making  AE  = 
2AD.    Then  AE  is  the  required  sect. 
Determine  the  accuracy  of  this  con- 
struction by  computing  the  ratio  of  AE  to  AB. 

Suggestion :  Let  r  =  radius.    Compute  AB  and  AE  in  terms  of  r,  then  divide. 


THE  CIRCLE  295 

1246.  Another  method  of  finding  the  approximate  value  of  the 
circle  is  as  follows:  Draw  diameter  CD.    Make  central  angle  AOB 
=  30°.  DrawA£_LCD.  Draw 

CE  tangent  at  C  and  equal 
to  3CZ>.  Draw  BE.  Then 
BE  equals  the  circle.  Deter- 
mine the  accuracy  of  this 
construction. 

1247.  In  making  a  drawing  for  an  arch  it  is  required  to  mark 
off  on  a  circle  drawn  with  a  radius  of  5  in.  an  arc  that  shall  be  8  in. 
long.    This  is  best  done  by  finding  the  angle  at  the  center.    How 
many  degrees  are  there  in  this  angle? 

1248.  The  perimeter  of  the  circumscribed  equilateral  triangle  is 
double  that  of  the  similar  inscribed  triangle. 

1249.  Squares  are  inscribed  in  two  circles  of  radii  2  in.  and  6  in., 
respectively.    Find  the  ratio  of  the  areas  of  the  squares,  and  also 
of  the  perimeters. 

1250.  Squares  are  inscribed  in  two  circles  of  radii  2  in.  and  8  in. 
respectively,   and  on  their  sides  equilateral  triangles  are  con- 
structed.   What  is  the  ratio  of  the  areas  of  these  triangles? 

1251.  A  log  a  foot  in  diameter  is  sawed  so  as  to  have  the  cross- 
section  the  largest  square  possible.    What  is  the  area  of  this  square? 
What  would  be  the  area  of  the  cross-section  of  the  square  beam  cut 
from  a  log  of  half  this  diameter? 

1252.  If  r  denotes  the  radius  of  a  regular  inscribed  polygon,  a 
its  apothem,  s  a  side,  A  an  angle,  and  C  the  angle  at  the  center, 
show  that: 

(a)  In  a  regular  inscribed  triangle  s=rV3,  a=|r,  A  =60°, 


(b)  In   a   regular   inscribed   quadrilateral  s=rA/2,  a== 
4  ==90°,  C  =  90°. 

(c)  In  a  regular  inscribed  hexagon  s^=r,  a=^-rv  3,  A  =  120°, 
C=60°. 

(d)  In     a    regular    inscribed    decagon    s=^r(  V  5  —  1),     a^ 

\r\10+2V5,  A  =  144°,  C  =  36°. 

1253.  If  r  is  the  radius  of  a  circle,  a  the  apothem  of  a  regular 
inscribed  n-gon,  and  in  one  of  its  sides,  i2n  a  side  of  a  regular  inscribed 


PLANE  GEOMETRY 


2n-gon,  cn  a  side  of  a  regular  circumscribed  n-gon;  An  the  area  of 
a  regular  inscribed  n-gon,  and  AZn  that  of  a  regular  inscribed 
2n-gon,  fill  out  the  accompanying  table: 


Given 

Required 

Given 

Required 

1.  r,  in 

a 

15.  in,  n  =  3 

An 

2.  r,  in 

fe. 

16.  r,  n  =  3 

An 

3.  r,  ijn 

in 

17.  in,  n=6 

An 

4.  r,  in 

Cn 

18.  r,  n  =  6 

An 

5.  r,  n  =  3 

in,  a 

19.  in,  n  =  12 

An 

6.  r,  n  =  6 

in,  a 

20.  r,  n  =  20 

An 

7.  r,  n  =  12 

in,  a 

21.  in,  n  =  12 

An 

8.  r,  n  =  4 

in,  a 

22.  r,  n  =  12 

An 

9.  r,  n=8 

in,  a 

23.  in,  n  =  4 

An 

10.  r,  n  =  10 

in,  a 

24,  r,  n  =  4 

An 

11.  r,  n  =  5 

in,  a 

25.  in,  n  =  8 

An 

12.  in,  n,  a 

An 

26.  r,  n  =  8 

An 

13.  in,  n,  r 

An 

27.  r,  n  =  5 

An 

14.  in,  n,  r 

Ain 

28.  r,  n  =  10 

An 

Theoretic 

1254.  The  area  of  a  regular  inscribed  hexagon  is  a  mean  propor- 
tional between  the  areas  of  the  inscribed  and  circumscribed  equi- 
lateral triangles. 

d!255.  An  equilateral  polygon  circumscribed  about  a  circle  is 
regular  if  the  number  of  its  sides  is  odd. 

d!256,  An  equiangular  polygon  inscribed  in  a  circle  is  regular  if 

the  number  of  its  sides  is  odd. 

1257.  If  C  be  a  point  in  the 
straight  line  AB,  the  three  semi- 
circles, drawn  respectively  upon 
sects  AB,  AC,  and  CB  as  dia- 
meters,  bound  an  area  equal  to  a 
circle  of  which  the  diameter  is 
the  perpendicular  CD,  D  being  in  the  largest  semicircle. 

1258.  If  upon  three  sides  of  a  right  tri- 
angle semicircles  be  drawn  as  indicated  in 
the  diagram,  the  area  of  the  right  triangle 
is  equal  to  the  sum  of  the  two  crescent- 
shaped    areas,    bounded    by  the     semi- 
circles.    (Hippocrates'  Theorem.)  A 

1259.  Give  a  simpler  proof  for  Ex.  859.    (b)  Generalize  this  fact. 


CHAPTER  VI 


METHODS  OF  PROOF 

There  are  two  general  methods  of  proving  theorems,  the  direct 
or  synthetic,  and  the  indirect  method.  Each  of  these  methods  of 
proof  may  be  in  its  nature  geometric  or  algebraic. 

Further,  indirect  proofs,  whether  geometric  or  algebraic,  may 
take  different  forms.  Thus  a  theorem  may  be  proved  indirectly, 
either  by  means  of  exclusion  or  reduction  to  an  absurdity,  or  by 
analysis. 

It  is  the  object  of  this  chapter  to  give  an  illustration  of  each 
of  these  methods  of  proof,  together  with  several  exercises  that  will 
be  most  naturally  proved  by  that  method. 

A.  THE  DIRECT  OR  SYNTHETIC  METHOD 

In  this  method  we  employ  either  superposition  or  start  with  the 
data,  and  combining  them  with  known  truths  proceed  step  by 
step  until  we  arrive  at  the  desired  conclusion. 

I.  GEOMETRIC  PROOF.*  B 

/ 

Illustration:  The  bisector  of  the  vertex 
angle   of    an    isosceles    triangle    bisects 
the  base. 
Given:    AABC,   AB^BC,  X  in  AC  so    that 


Prove:   AX=XC. 

In  &ABX  and  ACBX. 

(1)  AB=BC,  2S^ABX  = 

(2)  BX=BX. 

(3)  .'.  AABX^ACBX 

(A\    •    AY — r'Y 
\*)  '  •  <*"»  — ^>-A  • 


PROOF 


(1)  Data. 

(3)  Two   sides   and   the    included 
angle  determine  a  triangle. 

(4)  Horn,   sides   of    cong.   A    are 
equal. 


*  The  method  of  superposition  should  be  very  rarely  used.  For  illustra- 
tions of  it,  recall  the  proofs  of  the  fundamental  theorems  in  congruence  of 
triangles. 

297 


298 


PLANE  GEOMETRY 


EXERCISES.    SET  LXXXIX.     SYNTHETIC  METHODS  OF  PROOF 
Give  a  synthetic  proof  of  each  of  the  following: 

1260.  The  bisector  of  the  vertex  angle  of  an  isosceles  triangle 
is  perpendicular  to  the  base. 

1261.  If  the  perpendicular  bisector  of  the  base  of  a  triangle 
passes  through  the  vertex,  the  triangle  is  isosceles. 

1262.  Any  point  in  the  bisector  of  the  vertex  angle  of  an  isosceles 
triangle  is  equidistant  from  the  ends  of  the  base. 

II.  ALGEBRAIC  PROOF. 

Illustration:  If  one  leg  of  an  isosceles  triangle 
is  extended  through  the  vertex  by  its  own  length, 
the  sect  joining  its  end  to  the  end  of  the  base 
is  perpendicular  to  the  base. 
Given:  ABD  a  st.  line,  AB  =BD  =BC. 
Prove:  DCJ.AC. 

PROOF 

(1)  The  sum  of  the  4  of  a  A  is 
a  st.  4. 

(2)  Data. 

(3)  Base  2^ of anisosceles  Aareequal . 

(4)  Quantities  may  be  substituted 
for  their  equals  in  an  equation. 

(5)  The  whole  equals  the  sum  of  all 
its  parts. 

(6)  See  (4). 

(7)  Quotients  of  equals  divided  by 
equals  are  equal. 

(8)  The  numeric  measure  of  a  rt.  2$. 
is  90°. 

(9)  By  def.  of  J_. 


(1) 

(2) 

(3) 


=BD. 
4.DCB. 


(4)  .'.     substituting     (3)     in     (1) 


(5)  But 

(6) 
(7)  . 


(8)  .*.  %-DCA  is  a  rt. 

(9)  .*.  DCA.AC. 


This  method  is  especially  adapted  to  the  proof  of  numerical 
relations  between  angles  or  sects. 

The  following  procedure  is  generally  used  in  applying  the  alge- 
braic synthetic  type  of  proof. 

1.  Observation  of  the  numeric  relations  that  immediately  follow 
from  the  data. 

2.  Statement  of  these  relations  in  algebraic  form — the  equality 
or  the  inequality. 

3.  Reduction  of  these  algebraic  relations  by  the  help  of  axioms 
until  the  desired  conclusion  is  reached. 


METHODS  OF  PROOF  299 

EXERCISES.    SET  LXXXIX  (concluded) 

Give  an  algebraic  synthetic  proof  of  each  of  the  following : 

1263.  The  bisectors  of  two  supplementary  adjacent  angles  are 
perpendicular  to  each  other. 

1264.  If  the  bisectors  of  two  adjacent  angles  are  perpendicular 
to  each  other,  those  angles  are  supplementary. 

1265.  If  two  sides  of  a  triangle  are  unequal  the  angles  opposite 
them  are  unequal  in  the  same  order. 

1266.  The  sum  of  the  altitudes  of  a  triangle  is  less  than  its 
perimeter. 

1267.  The  angle  whose  sides  are  the  altitude  from  and  the  bisector 
of  an  angle  of  a  triangle  is  equal  to  one-half  the  difference  between 
the  remaining  angles  of  the  triangle. 

B.  THE  INDIRECT  METHODS 
I.  GEOMETRIC;  or  II.  ALGEBRAIC. 

a.  By  the  Method  of  Exclusion. 

Two  magnitudes  of  the  same  kind  may  bear  one  of  three  rela- 
tions to  each  other.  The  first  may  be  less  than,  equal  to,  or  greater 
than  the  second.  If  it  can  be  shown  that  two  of  these  relations 
are  false,  the  third  is  of  necessity  true.  Similarly  the  position  of  a 
point  may  be  fixed  by  the  method  of  exclusion. 

Illustration  1,  Theorem  21c:     If  two  angles  of  a  triangle  are 
unequal,  the  sides  opposite  them  are        B 
unequal  in  the  same  order. 

Given:    &ABC  in  which  %.A  >  %.C. 
Prove:   a>c. 

PROOF 


o>c,  a=c,  or  a<c. 

Suppose  a<c. 

Then  ^.A<^.C. 

But  this  contradicts  the  data. 

.-.  a<c.  _ 

Suppose  a=c. 

Then  %.A  =  4.C. 

But  this  contradicts  the  data. 

/.  a^_c. 

.*.  a>c. 


Authorities  left  to  the  student. 


300 


PLANE  GEOMETRY 


Illustration  2,  Theorem  54a,  Cor.  1: 
A  quadrilateral  whose  opposite  angles 
are  supplementary  is  inscriptible. 
-  Given:   Quadrilateral  ABCD  in  which  2LA 


Prove:   A,  B,  C,  D  coney  clic. 
PROOF 


(1)  A  circle  may  be  passed  through  A,  B,  C. 

(2)  D  lies  outside,  inside,  or  on  this  circle. 

(3)  Suppose    D  lies  inside    QABC  in  the 
position  Di. 

Then  ^.B^AXZ  and 
(4) 


Authorities  left  to  the  student. 


(5)  .*.  D  cannot  lie  inside  the  circle. 

(6)  Suppose   D  lies  outside  QABC  in  the 
position  I)j. 

Then  %.B  -°L^/Pfr  and  ^D^H  (CfiT-ZP). 

(7)  /.  2$.B  +  2H.D>*-M(APC+tfTA  -  ZP)  = 
H(360°-ZP)<180°. 

(8)  .*.  D  cannot  lie  outside  the  circle. 

(9)  .*.  D  lies  on  the  circle  and  ABCD  is  in- 
scriptible. 

EXERCISES.     SET  XC.    PROOF  BY  THE  METHOD  OF  EXCLUSION 
Prove  the  following  facts  by  means  of  the  method  of  exclusion  : 

1268.  Knowing  (1)  that  equal  chords  subtend  equal  arcs,  and 
(2)  that  unequal  chords  subtend  arcs  unequal  in  the  same  order, 
prove  the  converse  of  each  of  these  facts  by  the  method  of  exclusion. 

1269.  In  a  fashion  similar  to  that  used  to  prove  Ex.  1268,  show 
(1)  when  a  =  b,  c=d; 

that  if    (2)  when  a>b,  c>d;    then  the  converse  of  each  of  these 

(3)  when  a<b,  c<d, 
facts  is  true.    (This  is  known  as  the  Law  of  Converses.) 

b.  By  Reduction  to  an  Absurdity.     (Reductio  ad  absurdum.) 
This  method  is  similar  to  that  of  exclusion  in  that  it  makes  an 
assumption  which  results  in  a  contradiction  of  the  data,   but 
differs  from  it  in  that  but  one  such  assumption  is  made  before  a 
final  conclusion  is  reached.    Briefly,  in  proving  a  proposition  by 
reduction  to  an  absurdity,  we  do  so  simply  by  proving  that  the 
theorem  which  contradicts  the  conclusion  of  the  original*  is  false. 
*  Such  a  theorem  is  called  the  contradictory  of  the  direct. 


METHODS  OF  PROOF 


301 


Illustration:  If  the  median  to 
the  base  of  a  triangle  meets  it 
obliquely,  the  remaining  sides  are 
unequal. 

Given:  D  in  AC  in  AABC  ;  AD  =  DC, 

\ADB7*  ^.BDC. 
Prove:   AB^BC. 

PROOF 


D 


BC. 


Authorities  left  to  the  student. 


(1)  Suppose  AB 
Then 

(2)  .', 

(3)  But  this  contradicts  the  data  and 
.*.  the  assumption  is  absurd. 

(4)  /.  AB^BC. 

EXERCISES.   SETXCI.    PROOF  BY  REDUCTION  TO  AN  ABSURDITY 
Prove    the    following    exercises    by    the 
method  of  reduction  to  an  absurdity: 

1270.  Prove  that  if  two  angles  of  a  tri- 
angle are   equal,  the  sides  opposite  them 
are  equal. 

XX  Hint :  Suppose  A B  >  BC  and  take  AX=BC. 

1271.  If  upon  a  common  base  an  isosceles 
u  and  a  scalene  triangle  are  constructed,  the 

line  joining  their  vertices  does  not  bisect  the  vertex  angle  of  the 
isosceles  triangle. 

Hint:   Assume  that  it  does  bisect  it. 

1272.  If  perpendiculars  are  drawn  to  the  sides  of  an  acute  angle 
from  a  point  inside  the  angle,  they  enclose  an  oblique  angle. 

1273.  Prove  that  if  two  triangles  resting  on  a  common  base 
have  a  second  pair  of  sides  equal,  and  the  third  vertex  of  each 
outside  the  other  triangle,  their  third  sides  are  unequal. 

c.  By  the  Method  of  Analysis. 

The  method  of  analysis,  which  is  attributed  to  Plato,  was 
undoubtedly  used  by  Euclid,  but  was  probably  emphasized  by 
the  former. 

The  analysis  of  a  theorem  is  a  course  of  reasoning,  whether  con- 
scious or  unconscious,  by  means  of  which  a  proof  is  discovered. 


302  PLANE  GEOMETRY 

It  consists  of  discovering  the  immediate  condition  under  which  the 
conclusion  would  be  true,  and  continuing  to  do  this  with  each 
new  condition  until  one  known  to  be  true  is  reached. 

Analysis  always  takes  the  following  form:  Suppose  we  wish  to 
prove  that  A=B  if  C=D. 

We  start  by  saying:  A=B  if  x=y. 
But  x=  y  if  ra=n. 

and  m=  n  if  C=D. 

But  C=D  by  data. 

The  proofs  of  theorems  are  usually  put  in  the  synthetic  form, 
but  they  are  derived  analytically  and  then  rearranged  by  retracing 
the  steps  taken. 

The  analytic  method  might  also  be  called  the  method  of  reduc- 
tionj  or  of  successive  substitutions. 

If  we  wish  to  discover  how  to  prove  a  theorem,  we  should  always 
use  the  analytic  method.  It  is  much  more  likely  to  suggest  those 
helpful  auxiliary  lines  which  are  frequently  needed  before  any 
relation  between  the  theorem  to  be  proved  and  a  known  theorem 
is  apparent. 

On  the  contrary,  nothing  in  the  synthetic  method  suggests  the 
use  of  suitable  auxiliary  lines,  and  though  we  may  continue  to 
make  deductions  as  they  occur  to  us,  we  often  waste  time  and 
energy  without  getting  any  nearer  the  conclusion. 

Illustration  1,  Theorem  216:  If  two  sides  of  a  triangle  are  unequal, 
the  angles  opposite  them  are  unequal  in  the  same  order. 
Given:  a>c  in  AABC. 


Prove: 

Analysis  :    2$.  A  >  %.C  if  part  of  4.  A  >  4<7. 

Part  of  4.  A  >  4C  if  that  part  can 
be  made  an  exterior  4  of  a  A  in  which 
4C  is  a  non-adjacent  interior  4. 
This  cannot  be  done  as  the  4s  are  now  placed. 

How,  then,  can  we  find  an  4  which  is  equal  to  part  of  %-A  and  placed 
as  desired? 

Since  c<  a,  an  isos.  A  can  be  formed  by  laying  off  BX=c  on  a. 

Hence  the  auxiliary  line  AX  is  suggested,  and  %-A  >  4(7  if  ^.BAX>  4.C. 


But  XrBXA>XrC. 

:.  We  may  reverse  the  steps  and  give  a  brief  synthetic  proof  if  desired. 


METHODS  OF  PROOF  303 

The  analytic  method  does  not  always  lead  at  once  to  the  shortest 
proof  of  a  theorem,  though  it  is  far  more  likely  to  do  so  than  the 
synthetic  method.  At  times  an  analysis  suggests  several  methods 
of  proof,  and  our  selection  will  depend  upon  which  seems  the 
shortest.  Skill  in  selection  can  be  acquired  only  by  practice. 

Illustration  2:  If  one  median  in  a  triangle  is  intersected  by  a 
second,  the  sect  between  the  vertex  and  the  point  of  intersection 
is  double  the  other  sect. 

Given:  Medians  CD  and  AE  intersecting 

at  0  in  A  ABC. 

Prove:  AO=20E,  and  CO=20D. 
Analysis:  One  sect  may  be  proved  double 

another  by  proving  (1)  half  the  longer 

=the  shorter,  or  (2)  double  the  shorter      .  _ 

=the  longer.  A 

The  first  method  suggests  that  we  take  F  and  G  in  AO  and  CO  so  that 


_ 

Now  AO  =20E  and  CO  =20D  if  FO  =OE  and  GO  =OD. 
FO  =OE  and  GO  =OD  if  GD  and  FE  are  diagonals  of  a  CJ. 
FE  and  GD  are  diagonals  of^n^DE^FG  and  DE  \\  FG. 
DE  =  FG  and  DE  \\  FG  if  DE  bisects  AB  and  BC  in  AABC,  and  if 
FG  bisects  AO  and  CO  in  A  AOC,  for  then  DE=  \(AC)  =  FG  and  DE  \\ 
AC\\FG. 
.'  .  a  synthetic  proof  can  now  readily  be  given. 

Success  in  this  type  of  work  very  often  depends  on  the  selection 
of  suitable  auxiliary  lines.  Those  which  are  most  often  of  use  are 
discovered  by 

(a)  joining  two  points, 

(6)  drawing  a  line  parallel  to  a  given  line, 

(c)  drawing  a  line  perpendicular  to  a  given  line, 

(d)  bisecting  given  sects  as  in  the  last  illustration, 

(e)  producing  a  sect  by  its  own  length,  as  might  have  been  done 
in  the  last  illustration. 

Since  we  have  not  proved,  even  if  it  be  true,  that  the  algebraic 
processes  employed  in  the  proof  of  theorems  in  proportion  are 
reversible,  only  a  synthetic  proof  is  valid.  Of  course,  to  suggest 
the  synthetic  line  of  argument  it  is  desirable  to  give  an  analysis 
first  if  needed. 


304 


PLANE  GEOMETRY 


Illustration: 
c 
d' 

Prove:   ?-„  = 


Given:  -, 
b 


ANALYSIS 


02_02+C2 

(D 

Operation  perfor 

med. 

fo2       £)2_J_^2 

if  a262+a2d2; 

3a262+62c2 

(2) 

(1)  -62(62+d2) 

(2)  is  true 

if  a2d2  =  62c2 

(3) 

(2)   -a262 

(3)  is  true 

if  ad=bc 

(4) 

+  V(3) 

(4)  is  true 

.fa    c 

*i=d 

(5) 

(4)   +bd 

But  |=| 

by  data. 

.'  .  the  following  synthetic  proof 

a_c 

(D 

(1)  Given. 

(D  -M, 

:.ad=bc 

(2) 

(2)  Products  of 

equals  multiplied 

by  equals  are  equal 

(2)  ', 

/.  a2d2=62c2 

(3) 

(3)  Like  powers  of  equals  are  equal. 

(3)   +a26«, 

/.  a2d2+a252  = 

(4)  Sums  of  equals  added  to  equals 

62c2+a262 

(4) 

are  equal. 

fA\      •   K2/7»5 

+^1),...  g2=Q22+c2 

(5) 

(5)  Quotients  of 

equals  divided  by 

(4;     -rOf(& 

equals  are  equal. 

EXERCISES.    SET  XCII.    ANALYTIC  METHOD  OF  PROOF 

Give  an  analysis  of  each  of  the  following  exercises,  and  follow 
it  by  a  concise  synthetic  proof. 

1274.  Prove  the  second  illustra- 
tion under  the  analytic  method  by 

(a)  proving  ADOE^AFOG  or 
ADOF^&GOE  (p.  303). 

(6)  Doubling  DO  and  OE.  Use 
each  of  the  three  methods  sug- 
gested by  the  following  figures. 


Fia.  2 


Fio.  3 


METHODS  OF  PROOF  305 

1275.  Prove  Theorem  31,  Cor.  4,  analytically. 

1276.  Prove  Theorem,  31  Cor.  5,  analytically. 

1277.  If  one  acute  angle  of  a  right  triangle  is  double  the  other, 
the  shorter  leg  is  one-half  the  hypotenuse.    Prove  first  by  drawing 
auxiliary  lines  outside  the  triangle,  and  then  by  drawing  them 
inside  the  triangle. 

1278.  A  median  of  a  triangle  is  less  than  half  the  sum  of  the 
adjacent  sides. 

1279.  Prove  the  theorem  given  under  A  II  analytically  (p.  298)  . 

1280.  Prove  the  theorem  given  under  B  I  a  analytically  (p.  299). 

1281.  Prove  Ex.  1270  analytically. 

1282.  If 

ti    * 
prove  that  r=  -j. 

o     a 


1284.  If  J  4  prove  that 

19RR    Tfa    c  fh      a2_ 

1285.  If  r  =  j,  then  -  1  —  r^  —  =  —  >  —  -J 

6    d  a3  -63  c3  -d 

a  c 

While  practice  alone  can  give  skill  in  the  proof  of  theorems,  the 
following  suggestions  may  be  of  help  to  the  student. 

First.    Make  as  general  a  figure  as  possible. 

If  a  fact  is  to  be  proved  concerning  triangles  in  general  the 
figure  should  be  that  of  a  scalene  triangle,  since  many  facts  are 
true  of  isosceles  or  equilateral  that  are  not  true  of  scalene  triangles. 
Again,  if  a  fact  is  to  be  proved  concerning  quadrilaterals  in  general, 
it  might  be  misleading  to  draw  a  parallelogram  or  even  a  trapezoid. 

Second.  Always  have  clearly  in  mind  what  is  given  and  what  is  to 
be  proved. 

Third.    If  the  proof  is  not  readily  seen,  resort  to  analysis. 

Fourth.  Give  a  proof  by  the  method  of  reduction  to  an  absurdity 
only  as  a  last  resort. 


20 


CHAPTER  VII 

CONSTRUCTIONS.    METHODS  OF  ATTACKING 
PROBLEMS 

The  solutions  of  the  following  fourteen  problems  and  the  corol- 
laries to  them  are  typical  of  one  class  of  solution  of  construction 
problems,  namely,  those  solutions  which  are  at  once  found  to  rest 
directly  upon  some  known  theorem,  and  in  addition,  at  times,  upon 
some  known  construction. 

Problem  1.*  Draw  a  perpendicular  to  a  given  line  from  a  given 
point  (a)  outside  the  line,  (b)  on  the  line.] 

The  constructions  rest  directly  upon  Theorem  40,  Cor.  1,  the 
fact  that  two  points  equidistant  from  the  ends  of  a  sect  determine 
the  perpendicular  bisector  of  that  sect. 

Problem  2.    Bisect  a  given  (a)  sect,  (b)  angle,  (c)  arc. 

The  construction  of  (a)  rests  immediately  upon  Problem  1. 

The  construction  of  (6)  rests  immediately  upon  Theorem  5, 
the  fact  that  three  sides  determine  a  triangle. 

The  construction  of  (c)  rests  immediately  upon  Theorem  43, 
the  fact  that  equal  central  angles  intercept  equal  arcs,  and 
Problem  2  (6). 

Problem  3.    Reproduce  a  given  angle. 

The  construction  rests  directly  upon  Theorem  5. 

Problem  4.  Draw  a  line  through  a  given  point,  and  parallel  to  a 
given  line. 

The  construction  may  rest  directly  upon  Theorem  11,  the  fact 
that  if  when  lines  are  cut  by  a  transversal  the  alternate-interior 
angles  are  equal,  the  lines  thus  cut  are  parallel,  and  Problem  3. 

*  The  first  thirteen  problems  of  the  syllabus  were  taken  up  as  exercises 
in  the  First  Study,  but  are  repeated  here  (with  only  suggestions  for  their  con- 
struction) as  an  integral  part  of  a  syllabus  of  constructions. 

t  According  to  Proclus,  this  problem  was  first  investigated  by  (Enopides, 
a  Greek  philosopher  and  mathematician  of  the  5th  century  B.C.     Proclus 
speaks  of  such  a  line  as  a  "gnomon" — the  common  name  for  the  vertical  piece 
on  a  sundial. 
306 


METHODS  OF  ATTACKING  PROBLEMS  307 

Problem  5.  Construct  a  triangle,  given  any  three  independent 
parts;  (a)  two  angles  and  the  included  side,  (b)  two  sides  and  the 
included  angle,  (c)  three  sides,  (d)  the  hypotenuse  and  a  leg  of  a 
right  triangle. 

(What  can  you  say  in  case  (c)  if  one  side  is  equal  to  or  greater 
than  the  sum  of  the  other  two  sides?) 

Problem  6.    Divide  a  sect  into  n  equal  parts. 

The  construction  may  rest  upon  Theorem  31,  Cor.  3,  the  fact 
that  parallels  which  intercept  equal  sects  on  one  transversal  do 
so  on  all  transversals,  and  Problem  3.  For  further  help  see  Ex.  429, 
p.  116. 

Problem  7.  Find  a  common  measure  of  two  commensurable 
sects. 

Given:  ~KB  and  CD  commensur-  4r  -^ 

able  sects.  ~ 

Required  :  A  common  measure  ra.  C  -  «  -  £  -  D 
Solution:   Lay  off  A  7=  CD  on  AB. 


Lay  off  CZ=2YB  on  CD. 

Layoff  YX='ZD  on  YB. 

Lay  off  XB  on  ZD. 

It  is  found  to  be  contained  exactly  in  ZD. 

Then  XB  is  the  greatest  common  measure  of  AB  and  CD,  and  any 
integral  part  of  XB  is  a  common  measure  of  them.    Prove  it. 

Problem  8.    Pass  a  circle  through  three  non-collinear  points. 
The  construction  rests  directly  upon  Theorem  416. 
Cor.  1.    Circumscribe  a  circle  about  a  triangle.* 

Problem  9.  Divide  a  given  sect  into  parts  proportional  to  n 
given  sects. 

The  construction  rests  directly  upon  Theorem  31,  Cor.  2. 

Problem  10.  Divide  a  given  sect  harmonically  in  the  ratio  of 
two  given  sects. 

Use  (1)  the  method  suggested  in  Problem  9,  or  (2)  the  method 
suggested  by  Theorem  326,  Cor.  1. 

*  The  center  of  a  circle  circumscribed  about  a  polygon  is  called  its  circum- 
center. 


308  PLANE  GEOMETRY 

Problem  11.    Find  a  fourth  proportional  to  three  given  sects. 
The  construction  rests  directly  upon  Theorem  31.    The  solution 
is  given,  but  the  proof  is  left  to  the  pupil. 

a  e  Given:    Sects  a,  b,  c. 

Required:  Sect  x,  so  that  ~== 
&  6    d' 


Solution:   Construct  any  angle  RST. 
x-       -N  On  ST,  lay  off  SV=a. 

^  \  On  SR,  lay  off  SW  m  b. 

/"  \  On  FT,  lay  off  VQ=c. 

;\  \  DrawFF. 

,'       \  \  Draw  QY  \\  VW  and  cutting  SR 

'— Tf-V x Vfe     at  — 

Then  WY  is  the  required  sect  x. 

Cor.  1.    Find  a  third  proportional  to  two  given  sects. 

What  is  the  only  modification  needed  in  the  construction  of 

Problem  11  in  order  to  find  x,  so  that  4-  =  —  ? 

b       x 

Problem  12.  Upon  a  given  sect  as  homologous  to  a  designated 
side  of  a  given  polygon  construct  another  similar  to  the  original 
polygon. 

The  construction  rests  directly  upon  Theorem  366  and 
Problem  11. 

Problem  13.  Construct  a  square  equal  to  the  sum  of  two  or 
more  given  squares. 

Use  the  Pythagorean  Theorem. 

Cor.  1.  Construct  a  square  equal  to  the  difference  of  two  given 
squares. 

In  this  case  the  larger  of  the  given  squares  will  be  the  square 
on  which  side  of  the  right  triangle? 

The  construction  therefore  rests  upon  Problem  5  (d) 

Cor.  2.  Construct  a  polygon  similar  to  two  given  similar  poly- 
gons and  equal  to  (a)  their  sum,  (b)  their  difference. 

How  do  the  areas  of  similar  polygons  compare? 

Problem  14.  Inscribe  in  a  circle,  regular  polygons  the  number 
of  whose  sides  is  (a)  3  *2n,  (b)  4  -2". 

(a)  The  construction  rests  directly  upon  Theorem  58  and  the 
construction  of  an  equilateral  triangle  given  its  side  (here  the 
radius  of  the  circle)  in  order  to  obtain  a  central  angle  of  120°. 

(b)  What  is  the  central  angle  in  the  case  of  the  square? 


METHODS  OF  ATTACKING  PROBLEMS  309 

THE  SYNTHETIC  METHOD  OF  ATTACKING  A  PROBLEM 

The  preceding  type  of  construction  problem  is  the  simplest,  and 
the  solution  of  one  of  that  nature  is  usually  so  readily  seen,  that 
without  further  explanation  (except  for  one  or  two  suggestions 
appended  to  the  first  exercises)  the  pupil  is  asked  to  solve  the  fol- 
lowing set  of  exercises. 

EXERCISES.    SET  XCIII.    SYNTHETIC  SOLUTIONS 

1286.  Trisect  a  right  angle.     (We  know  that  each  angle  of  an 
equilateral  triangle  is  two-thirds  of  a  right  angle.    What  construc- 
tion does  this  therefore  suggest?) 

1287.  Divide  an  equilateral  triangle  into  three  congruent  tri- 
angles.    (We  know  that  the  bisectors  of  the  angles  of  a  triangle 
are  concurrent,  and  that  triangles  are  determined  by  two  angles 
and  the  included  side.) 

1288.  Construct  an  equilateral  triangle  with  a  given  sect  as 
altitude.    (What  fact  about  the  altitude  of  an  equilateral  triangle 
suggests  the  construction?) 

1289.  Construct  a  square  having  given  its  diagonal. 

1290.  Through  two  given  points  draw  straight  lines  which  shall 
make  an  equilateral  triangle  with  a  given  straight  line. 

1291.  On  a  given  sect  construct  a  rhombus  having  each  of  one 
pair  of  opposite  angles  double  each  of  the  other  pair. 

1292.  On  a  given  base  construct  a  rectangle  equal  to  (a)  a  given 
square,  (b)  another  given  rectangle,  (c)  a  given  triangle,  (d)  a 
given  trapezoid. 

1293.  The  sides  of  a  polygon  are  5,  7,  9,  11,  13.    Construct  one 
similar  to  it  having  the  ratio  of  similitude  3  to  5. 

1294    Construct  a  polygon  similar  to  the  accompanying  polygon, 
having  the  ratio  of  simili- 
tude equal  to  that  of  the  two 
given  sects  a  and  b. 

1295.  Construct  a  poly- 
gon similar  to  two  (or  more) 
given  similar  polygons  and 
equivalent  to  their  sum  (or  difference). 


310 


PLANE  GEOMETRY 


/ 


\ 


1296.  Construct  a  circle  equal  to  the  sum  of  two  given  circles. 

1297.  Construct  a  circle  equal  to          £ —     <^ 

the  difference  of  two  given  circles. 

Problem  15.  Construct  a 
triangle  given  two  sides  and 
the  angle  opposite  one  of  them. 
Given:   Sides  a  and  c]  2^.  A. 
Required:    AABC. 
Construction:  Construct 2^.PAQ= %.A. 
OnAQ,  layoff  AB=cL 

With  B  as  center  and  a  as  radius,  strike  an  arc  cutting  AP  in  C  and  Ci, 
touching  it  in  only  C ,  or  not  at  all,  according  asa>hb,a=hb,  or  a<hb. 
Then  AABC  and  AABCi  fulfill  the  required  conditions. 

Discussion:  I.  Conditions  under  which 
there  are  two  solutions. 

(1)  4.A  acute,  a  <c,  but>fo,  the 

perpendicular  from  B  to  A  P. 
II.  Conditions  under  which  there  is 
but  one  solution. 
(1)   2$_A  acute,  a=/i&  or  c. 

acute,  right,  or  obtuse, 
and  a>c. 


A  /  x^     A 

III.  Conditions  under  which 
there  is  no  solution.  , 

(1)  2$.  A    right    or  obtuse  °|       \f 
and  a<c. 

(2)  2$. A  acute  and  a  <hb.  A 

THE  DISCUSSION  OF  A  PROBLEM 

As  in  the  case  of  Problem  15,  many  exercises  in  construction 
call  for  a  discussion,  by  reason  of  the  fact  that  the  number  of  solu- 
tions of  the  problem  varies  under  different  conditions. 


METHODS  OF  ATTACKING  PROBLEMS          311 

It  will  be  noticed  that  the  discussion  of  Problem  15  has  been 
arranged  under  three  heads,  based  upon  the  number  of  possible 
solutions.  The  discussion  might  have  been  arranged  under  entirely 
different  headings.  To  show  this,  the  pupil  is  asked  to  fill  in  the 
discussion  under  the  following  heads. 
I.  <£A  acute.  II.  ^A  right.  III.  -£A  obtuse. 

(1)  a<hb.  (1)  a<c.  (1)  a<c. 

(2)  a=hb.  (2)  a=c.  (2)  a=c. 

(3)  a>hbbut  <c.        (3)  a>c.  (3)  a>c. 

(4)  CIEEEC. 

(5)  a>c. 
or, 

I.  a  <c.     II.  a=c.     III.  a>c.     (Fill  in  all  the  necessary  subheads.) 
Further  illustration  of  what  is  meant  by  the  discussion  of  a  problem 
in  construction:  To  find  a  point  X  which  shall  be  equidistant  from 
points  Pi  and  P2  and  at  a  given  distance  d  from  P3. 

Given:    Points  PI,  P2,  P3;  sect  d.  d 

Required:  Point  X  such  that  PiX  = 

Construction:    (1)  Draw  PiP2  and     Q  v     —    —     *- 

bisectitatA.  ^ ffl £^~j£^X$l — •»  . 

(2)  Erect  QARl.PJ^.  /  \ 

(3)  With  P3  as  center  and  d  \ p        I  \ 
as  radius  describe  a  circle  cut-                      a        /                 }jp>  1 


ting  QR  at  X,  -X"i. 
Then  X,  Xi  are  the  required  points. 

PROOF 


(1) 
and 
V  X, 


(2)  v  X  and  Xi  are  on  ©Pi, 


(1)  The  locus  of  points  equidistant 
from  the  ends  of  a  sect  is  the  perpen- 
dicular bisector  of  that  sect. 

(2)  Const.,  and  the  locus  of  points 
at  a  given  distance  from  a  given  point 
is  a  circle  of  which  the  center  is  the 
given  point,  and  the  radius  the  given 
distance. 

Discussion:   Draw  PzN A.QR- 

I.  Two  solutions:   If  d>P3N,  for  then  a  portion  of  QR  will  be  a  chord 
of  the  circle  P3. 

II.  One  solution:   If  d  =  P3N,  for  then  QR  will  be  tangent  to  OP3  at  X. 

III.  No  solution:   If  d<P3N,  for  then  all  points  in  QR  will  lie  further 
from  the  center  P3  than  the  length  of  the  radius. 


312  PLANE  GEOMETRY 

LOCI  AND  PROBLEMS  SOLVED  BY  THE  METHOD  OF 
THE  INTERSECTION  OF  LOCI 

In  many  problems  we  are  asked  to  find  the  position  of  a  point 
which  satisfies  two  given  conditions.  Each  of  the  conditions 
determines  a  locus  on  which  the  point  lies,  and  the  solution  of 
the  problem  is  therefore  the  point  or  points  common  to  both  loci. 

The  last  exercise  illustrates  this  type  of  problem.  The  points 
X  and  Xi  fulfill  two  conditions.  The  first  condition  determined 
the  locus  of  points  equidistant  from  PI  and  P2,  the  second  deter- 
mined the  locus  of  points  at  a  given  distance  d  from  P3.  The  solu- 
tion of  the  problem  was  those  points  common  to  both  loci. 

EXERCISES.    SET  XCIV.    INTERSECTION  OF  LOCI 
Give  a  full  discussion  of  each  of  the  exercises  in  this  set. 
To  find  a  point  X  such  that : 

1298.  X  shall  be  at  the  distance  d\  from  point  PI,  and  dz  from 
point  PZ. 

1299.  X  shall  be  equidistant  from  two  parallel  lines  and  at  the 
distance  d  from  point  P. 

1300.  X  shall  be  equidistant  from  two  intersecting  lines,  and 
also  equidistant  from  points  P  and  PI. 

In  line  I  find  point  X  so  that : 

1301.  X  shall  be  at  distance  d  from  P. 

1302.  X  shall  be  equidistant  from  P  and  PI. 

1303.  X  shall  be  at  distance  d  from  Zi. 

1304.  X  shall  be  equidistant  from  two  given  parallel  lines. 

1305.  Draw  a  circle  of  given  radius  to  touch  two  given  lines. 

Draw  a  circle  with  a  given  radius : 

1306.  Passing  through  two  given  points. 

1307.  Passing  through  a  given  point  and  touching  a  given  line. 

1308.  Passing  through  a  given  point  and  touching  a  given  circle. 

1309.  Touching  a  given  line  and  a  given  circle. 

1310.  Touching  two  given  circles. 

1311.  Describe  a  circle  touching  two  parallel  lines  and  passing 
through  a  given  point. 

1312.  Construct  a  right  triangle,  having  given  the  hypotenuse 
and  the  altitude  on  the  hypotenuse. 


METHODS  OF  ATTACKING  PROBLEMS 


313 


1313.  Construct  a  right  triangle,   given  sects  of  hypotenuse 
made  by  the  bisector  of  the  right  angle. 

1314.  Construct  a  triangle,  given  an  altitude  and  the  sects  made 
by  the  altitude  upon  the  opposite  side. 

1315.  Construct  a  right  triangle,  given  the  sects  of  the  hypo- 
tenuse made  by  the  altitude  upon  the  hypotenuse. 

Problem  16.    Through  a  given  point  draw  a  tangent  to  a  given 
circle,   (a)  when  the 

point  is  on  the  circle,  — ^^£L Q 

(b)  when  the  point 
is  outside  the  circle. 

Given:  PL  P  (a)  on  ©C, 

(6)  outside  QC. 
Required:  PQ  tangent  to 

OC. 
Construction :__ For  (a)  For  (6) 

(1)  Draw  CP.  (1)  Draw  sect  CP. 

(2)  Draw  PQ  J.CP.  (2)  On  CP  as  diameter  describeQO 
Then  PQ  is  the  required  tangent.         cutting  OC  in  Q  and  Qi. 

(3)  Draw   PQ   and   PQi,   each  of 
j     which  is  the  required  tangent. 

PROOFS 

For  (6) 
Draw  CQ  and  C^i. 

(1)  Then   A  PCQ  and  PCQi  are 
right  A.    Why? 

(2)  /.PQ   and  PQiJ_CQ  and  CQi 
respectively.     Why? 

(3)  .'.  PQ  and  PQi  are  tangent  to 
GC.    Why? 

Discussion:   When  the  point  is  on  the  circle  there  is  but  one  solution.    Why? 
When  it  is  outside  the  circle  there  are  two  and  only  two  possible  solu- 
tions.   Why? 

The  solution  of  all  construction  problems  calls  for  at  least  four 
parts.    Namely: 

1st — The  statement  of  what  is  given. 

2nd — The  statement  of  what  is  required. 

3rd — The  construction  of  what  is  required. 

4th — The  proof  of  the  correctness  of  this  construction. 


For  (a) 

Since  %.CPQ  is  a  rt.  2$_,  and  VP  a 
radius,  PQ  is  tangent  to  QC.     Why? 


314  PLANE  GEOMETRY 

We  have  alread}r  seen  that  many  problems  call  for  a  discussion, 
and  in  many  of  the  problems  in  the  syllabus  an  analysis  will  be 
given. 

Omitting  data  and  what  is  required,  one  might  say  then  that 
the  systematic  solution  of  a  problem  consists  of  four  parts:  (1)  The 
analysis,  (2)  the  construction,  (3)  the  proof,  and  (4)  the  discussion. 

Problem  17.    Construct   a 
common  (a)  external  and  (b) 
internal  tangent  to  two  circles. 
~"^-^ ^__    P  Y     Given:    Qs  C  and 


n        i  „--— -  -          Ci  of  radii  r  and 

°l     I   ^-~~~~         X        ri. 

Required:       The 
common  (a)  ex- 
ternal tangents 
TN  and   TWi, 
and  (6)  internal  tangents  TN  and  TiNi. 

Analysis  of  (a) :   Suppose  one  of  the  required  tangents,  TN,  drawn. 
Then  if  r  ^n,  TNX  will  intersect  CCiY  at  P. 
Then 


But  r  and  n  are  known. 

/.to  find  P  divide  CCi  externally  in  the  ratio  of  -' 


/.  to  find  tangent  TN,  draw  PN  tangent  to  QCi  and  show  that  if  pro- 
duced it  will  be  tangent  to  O^- 

The  completion  of  the  problem  is  left  to  the  student,  who  should  discuss 
the  case  where  r=r\. 
Analysis  of  (b):   The  analysis  in  this  case  is  similar  to  that  in  case  (a)  except 

that  CCi  will  be  divided  internally  in  the  ratio  of  r  to  r\. 
Construction  and  proof  left  to  the  student. 

Problem  18.    Inscribe  a  circle  in  a  triangle* 

The  center  is  determined  by  the  intersection  of  what  two  loci? 

Cor.  1.    Find  the  centers  of  the  escribed  circles  of  a  triangle.^ 

The  excenters  are  determined  by  the  intersections  of  what  loci? 

*  The  center  of  the  circle  inscribed  in  a  polygon  is  called  the  incenter  of  the 
polygon. 

f  The  circles  which  are  tangent  to  one  side  of  a  polygon  and  to  the  two  consecu- 
tive sides  produced,  are  called  the  escribed  circles,  and  their  centers  are  called  the 
excenters  of  the  polygon. 


METHODS  OF  ATTACKING  PROBLEMS  315 

Problem  19.    Construct  upon  a  given  sect  the  segment  of  a 
circle  capable  of  containing  a  given  angle. 


Given:   Sect  AB;  ^.C. 

Required:   Segment  of  QO  resting  on  AB  as  chord  such  that  the  inscribed 

3_  =  4C. 

Analysis:    If  the  construction  were  complete  ABC  would  be  any  one  of  a  num- 
ber of  triangles  of  base  AB  and  a  vertex  %.C. 

The  circle  circumscribed  about  any  one  AABC  would  give  the  required 
locus. 

.'.  the  following: 
Construction:   Const.  2^.XCY  =  ^C. 

With  B,  any  convenient  pt.  in  CY  as  center,  and  radius  =  sect  AB, 
strike  an  arc  cutting  CX  in  A  . 

Circumscribe  a  circle  0  about  AABC. 
Then  segment  ABC  is  the  required  segment. 
(Proof  left  to  the  student.) 

Problem  20.    Construct  a  mean  proportional  between  two  given 
sects. 

Given:   Sects  a  and  6.  -  - 


Required:  Sect  c,  so  that  ==  =  •  ~  -  -  -  — 

c      b 

Hint:   Can  you  word  Proposition  39,  Cor.  4,  to  relate  to  the  sects  of  a  diameter 
of  a  circle? 

Extreme  arid  Mean  Ratio.    //  sect  AB  is  so  divided  by  the  pt.  P 


that  -  —  =  =,  the  sect  is  said  to  be  divided  in  extreme  and  mean 
PB    AB 

ratio.    That  is,  one  part  of  the  sect  is  a  mean  proportional  between 
-  -  -    the  entire  sect   and  the  other  part. 

This  division  of  a  sect  is  known  as  the 

"golden  section."    As  in  harmonic  division,  a  sect  may  be  divided 
internally  and  externally  into  extreme  and  mean  ratio. 


316 


PLANE  GEOMETRY 


Problem  21.    Divide  a  given  sect  into  mean  and  extreme  ratio* 
Given:   Sect  *. 

b 

s+b 
Analysis:  Suppose  the  construction  completed.  / 

_,,        .,.      s         a          ,  s        b  I  ^^  \\ 

Then  (1)     -  •  T— :  and  ^  a  TT-I.  N  i  -     >n  I  \ 


Required:  Sects  a  and  b  so  that  (1)  -  =-~  and  (2) 


. 
-a          6     s+b 

These  suggest     f 
a  tangent  as  the     i#  __  _£. 

mean      proper-    ~j 
tional  between  a 

secant  and  its  external  sect  or  the  leg  of  a  rt.  A  as  the  mean  proportional  be- 
tween the  hypotenuse  and  its  projection  upon  the  hypotenuse.  As  the  propor- 
tions now  stand  there  is  but  one  known  term. 

But  by  composition  we  get  (1)       lib?=  -  and  (2)  ^^=  J. 

S  Q  S  b 

This  suggests  s  as  the  tangent  to  a  circle,  «+a  as  the  entire  secant,  and  a  as 
its  external  sect  on  the  one  hand,  and  b  as  the  entire  secant  and  b—s  as  its 
external  sect  on  the  other  hand. 

The  most  natural  secant  to  select  is  that  which  passes  through  the  center. 

Hence  the  following: 
Construction:    (1)  Const.  O#  of  diameter  s  tangent  to  XY.  (XY=s)  at  Y. 

(2)  Draw  secant  XQOP. 

(3)  On  XY  lay  off  XQ^XQ,  and  on  YX  produced  lay  off  XP^XP. 

(4)  Then  Qi  and  PI  are  the  required  points.    (Proof  left  to  the  student.) 
Problem  22.    Inscribe  in  a  circle  a  regular  decagon. 

Given:    ©0. 

Required:  AB,  the  side  of  the  regular  in- 

scribed decagon. 
Analysis:    If    AB=side    of    reg.    decagon, 


•I  OfJ°  _ 


=  72 


and  if  PB  bisects  4.  ABO,  %.PBO  =36°  = 

40  and  2piP5  =  1800-(360+72°)=720. 

/.  AAPB«>  AABO    and     AOPB     is 


PB=OP. 


AP     AB 


.  /.to  find  OP=AB   divide   CM,  the  radius  of  QO,  into 

C/P     C/^4. 
extreme  and  mean  ratio,  and  use  the  mean  as  AB.    (Proof  left  to  student.) 

*  A  painting  is  said  to  be  most  artistically  arranged  when  its  center  of  inter- 
est is  so  placed  that  it  divides  the  width  of  the  picture  into  extreme  and  mean 
ratio.  If  the  student  is  especially  interested  in  the  topic,  he  will  find  references 
in  Chapter  X. 


METHODS  OF  ATTACKING  PROBLEMS 


317 


Cor.  1.    Inscribe  in  a  circle  regular  polygons  the  number  of 
whose  sides  is  (1)  5.2",  (2)  15.2". 

(1)  Left  wholly  to  the  student. 
*  (2)  Hint:  ^i-TV 
To   transform   a  polygon  means  to 
change  its  shape  but  not  its         C* 
area. 

Problem  23.  Transform  a 
polygon  into  a  triangle. 

Given:  Polygon  ABCDEF. 
Required:  A  A=  ABCDEF. 
Hint:  With  CA  as  base,  what  is 

the  locus  of  the  vertices  of  triangles  whose  area  =  area  AABCt 

To  eliminate  side  A  B,  by  what  particular 

A  shall  we  then  replace  AABCt 
Similarly,  what  A  is  to  replace  ADEFf 
Similarly,  how  dispose  of  side  DH? 

Problem  24.  Construct  the  square 
equal  to  a  given  (a)  parallelogram,  (b) 
triangle,  (c)  polygon. 

Problem  25.    Construct  a  parallelogram  equal  to  a  given  square 
having  given  (a)  the  sum  of  base  and  altitude  equal  to  a  given  sect, 
(b)  the  difference  of  base  and  altitude  equal  to  a  given  sect. 
Given:     Square  S; 

sect  a. 
Required:   Ad =S  /  ^i      \ 

with  (a)  base  +  s  / 

alt.  =a,  (b)  base 


\ 


i  j  *  *-v  , 

— alt.  =a.  j 

(a)  From  Fig.  1  show  that  any  parallelo- 
gram with  QR  as  base  and  PQ  as  altitude  = 
square  S,  and  that  it  also  f  ulfills  the  second 
condition. 

(6)  From  Fig.  2  show  that  any  parallelo- 
gram with  QR  as  base  and  PQ  as  altitude  = 
square  s,  and  that  it  also  fulfills  the  second 
condition. 

This  problem  solves  geometrically  the 
algebraic  problems,  given  (a)  x+y  =  a 
and  xy=s,  (6)  x—y=a  and  xy  =  s,  find  x 
and  y. 


FIG. 


/ 


Q 


FIQ.  2 


318  PLANE  GEOMETRY 

Problem  26.    Construct  a  polygon  similar  to  one  and  equal  to 
another  of  two  given  polygons. 

Given:  Polygons  P  and  Q. 
.---'\  >/^^v  Required:    Polygon  R  so 

p      \\         R      \     \  Q         \     Analysis:    IfjRcopandz 

\  N     \  \        homologous  to  s,  then 

V  X  \  T-\  n 


If  P  and  Q  equal  ra2  and  n2  respectively,  and  Q  =R,  then  —  =  — . 
P    m'       ,  P     s2  n*      *' 

VQ^and]T:F2 

•"•  —  s— ,  /.  x  is  the  fourth  proportional  to  the  side  of  a  square  =P,  the 

side  of  a  square  =Q,  and  the  side  in  P  homologous  to  x. 
(Construction  and  proof  left  to  the  student.) 

Problem  27.    Construct  a  square  which  shall  have  a  given  ratio 
to  a  given  square.  D  ^ 

Given:   Square  S  of  side  s,  ratio  - 


E 


Required:     Square  R  of  side  r  so  that  Nv 

H- 

o      q  \ 

Analysis:  If  a  rectangle  were  to  be  found  \ 

,.    A  rect.  AE 
so  that  - 

square 

square  S  would  simply  have  to  be  divided  so  that  alt-  of  rect-  =  P. 

s  q 

Then  it  remains  to  convert  rect.  AE  into  a  square. 

(Construction  and  proof  left  to  the  student.) 

Cor.  1.*    Construct  a  polygon  equal  to  any  part  of  a  given 
polygon  and  similar  to  it. 

(Construction  left  to  the  student.) 

FORMAL  ANALYSIS  OF  A  PROBLEM 

When  problems  .(as  is  usually  the  case)  are  of  such  nature  that 
the  application  of  known  theorems  and  problems  to  their  solution 
is  not  at  once  apparent,  the  only  way  to  attack  them  is  by  a  method 

*  This  corollary  is  included  in  the  syllabus  because  of  its  practical  value  in 
drafting,  since  it  is  by  this  method  that  the  draftsman  finds  his  scale,  in  enlarg- 
ing or  reducing  a  diagram  of  any  sort. 


METHODS  OF  ATTACKING  PROBLEMS          319 

resembling  the  analytic  method  of  proof  of  a  theorem.     This 
method  is  called  the  analysis  of  the  problem. 

In  brief,  the  directions  for  following  this  method  are  : 

1.  Suppose  the  construction  completed. 

2.  Draw  a  figure  showing  all  the  parts  concerned.    (Given  parts 
in  heavy  lines,  and  required  parts  in  dotted  lines.) 

3.  Study  the  relations  of  the  parts,  and  try  to  find  some  relation 
which  will  suggest  a  possible  construction. 

4.  If  a  first  attempt  fails,  introduce  new  relations  by  means  of 
auxiliary  lines,  and  continue  the  study  of  relations  until  a  clue  to 
the  correct  construction  is  derived. 

5.  Look  for  that  clue  in  a  rigid  part  of  the  figure  —  usually  a 
triangle. 

A  few  more  illustrations  to  show  just  what  is  meant  by  these 
directions  will  be  helpful. 

1.  Construct  a  triangle,  having  given  the  base,  a  base  angle,  and 
the  sum  of  the  remaining  sides. 
Suppose  ABC  to  be  the  completed  triangle. 
In  it  we  know  AC=b,  ^A,  and  c+a. 
.'.produce  AB  to  X  so  that  BX  =  a. 
Then  ABCX  is  an  isos.  A. 
But    AAXC   is   determined    (6, 


But  ^XCB=  $X  V  BX=BC. 
.*.  AABC  is  determined. 

2.  Construct  a  triangle,  having  given  one  side,  the  median  to  that 
side,  and  the  altitude  to  a  second  side. 
Given:   b,  mb,  hc. 
Analysis:  Suppose  ABC  to  be  the  required  A. 

Then  the  known  parts  are  AC  .  _  5  _  B 


_ 

:.  AE  =EC  =*!.  CD  =h«.  BE  =               h0  D<Q/    \ 

*  S**\J       \ 

nth,  and  2$.  at  D^rt.  Xp.  ,''       \        \ 

ADAC  is  determined.    Why?  ''       ft      \    \ 

Using  this  triangle  as  the  basis  of  con-  /              /            x\  \ 

struction,  what  means  have  we  of  filing  the  /                 ''  _  ]j 
vertex  Bl                                                           A         b/2       $        *>A 


320 


PLANE  GEOMETRY 


8.  Construct  a  triangle  having  given  its  medians. 

_  Given:   ma,  ffib,  Me. 

J?  Required:    AABC. 

,'  /    \  Analysis:    Suppose 


.m0 


ABC 

to  be  the  required  A. 

Evidently  no  3. 
within  the  A  is  fixed 
by  the  medians. 

But  it  is  known 
that  OC  =  %me,  OEa 
and 


».„.     / 


that  OE  bisects  AC. 

.'.  if  OE  is  produced  to  X,  so  that  XE=EO,  a  parallelogram  is  deter- 
mined whose  sides  and  one  diagonal  are  known. 

Hence  the  following: 
Construction:    (1)  Trisect  ma,  mb,  me. 

(2)  Construct  ACXO  with  %mc,  %ma,  and  2/3mb  as  ita  sides. 

(3)  Complete  the  EJOCXA. 

(4)  Produce  EO  by  %mi>,  and  vertex  B  is  determined. 

4.  Construct  a  trapezoid  given  the  bases  and  the  base  angles. 
Analysis:   Suppose  ABCD  to  be  the  com- 
pleted trapezoid. 

We  know  AB,  DC,  2$.D,  and  %.C. 
Is,  then,  any  part  of  the  figure  de- 
termined? 

If  we  draw  BDl  ||  AD,  then  ABD& 
will  be  a  parallelogram  and  DDi  =AB. 

%.BDiC  =  2$.ADC  and  D^C^DC-AB  and  ABD,C  is  thus  determined. 
(Construction,  proof  and  discussion  are  left  to  the  student.) 
The  preceding  exercises  are  illustrative  of  analysis  geometric  in 
form,  but  the  analysis  of  a  problem  in  construction  may  also  be 
algebraic  in  form,  as,  for  example,  in  Problems  21,  26,  and  27. 

NOTE  1.  —  These  diagrams  show  the  symbolism  used  in  reference  to  tri- 
angles and  trapezoids  throughout  this  text. 


-4 


A 


A 


METHODS  OF  ATTACKING  PROBLEMS          321 


NOTE  2. — Construction  of  trapezoids. 

The  relations  given  by  the  following  constructions  are  often 
useful.  Let  ABCD  be  any  trapezoid.  Draw  CF  and  BH  \\  AD, 
CG  and  AH  ||  DB.  Join  H  (the  in- 
tersection of  AH  and  BH)  with  F, 
C,  G.  The  figures  ACGH,  ADBH, 
FCBH,  are  parallelograms.  BF=a 

W  -  B, 

180°- 


1318.  a,  hat  ma. 
1321.  6,  c,  h* 
1324.  ft.,  ho, 


EXERCISES.    SET  XCV.    PROBLEMS  CALLING  FOR  ANALYSIS 

Construct  a  triangle,  having  given : 
1316.  a,  b,  mb.          1317.  a,  b,  hb. 
1319.  a,  ma,  <£#.      1320.  ma,  ha, 
1322.  a,  fta,  hc.          1323.  ft.,  ££ 
1326.  a,  6,  %A  +  3:B. 

1326.  The  base,  the  altitude,  and  an  angle  at  the  base. 

1327.  The  base,  the  altitude,  and  the  angle  at  the  vertex. 
Construct  an  isosceles  triangle,  having  given: 

1328.  The  base  and  the  angle  at  the  vertex. 

1329.  The  base  and  the  radius  of  the  circumscribed  circle. 

1330.  The  base  and  the  radius 
of  the  inscribed  circle. 

1331.  The  perimeter  and  the 
altitude. 

Let  ABC  be  the  required  A, 


E  ~A D B  J 

EF  the  given  perimeter.  The  alti- 
tude CD  passes  through  the  mid- 
point of  EF,  and  the  As  EAC,  BFC 
are  isosceles. 

1332.  Construct  a  triangle,  having 
given  two  angles  and  the  sum  of  A 
two  sides. 

Can  the  third  <£  be  found?    Assume  the  problem  solved.    If 
AX  =  AB+BC,  what  kind  of  triangle  is  ABXCt     What  does 
equal?    Is  <&X  known?    How  can  C  be  fixed? 


B 


322  PLANE  GEOMETRY 

1333.  Construct  a  triangle,  having  given  a  side,  an  adjacent 
angle,  and  the  difference  of  the  other  sides. 

If  AB,  <&A,  and  AC  -BC  are  known, 
/'\       what  points  are  determined?    Then  can 
/''        \      XB  be  drawn?    What  kind  of  triangle  is 
I      AXBCt    How  can  C  be  located? 
I         1334.  Construct  an  isosceles  triangle, 
±J     having  given  the  sum  of  the  base  and 
an  arm,  and  a  base  angle. 

1335.  Construct  a  triangle,  having  given  the  base,  the  sum  of 
the  other  two  sides,  and  the  angle  included  by  them. 

1336.  Construct  a  triangle  given  the  mid-points  of  its  sides. 

1337.  Draw  between  two  sides  of  a 
triangle  a  sect  parallel  to  the  third  side, 
and  equal  to  a  given  sect. 

If  PQ  =  d,  what  does  AR  equal?    How     

will  you  reverse  the  reasoning?  A  R  B 

1338.  Draw  through  a  given  point  P  between  the  sides  of  an 

angle  A  OB  a  sect  terminated  by  the 
sides  of  the  angle  and  bisected  at  P. 
If  PM  =  PN,  and  PR  \\  AO,  what 
can  you  say  about  OR  and  RNt  Can 
you  now  reverse  this?  Similarly,  if 
PQ  \\BO,  isOQ  =  QM? 

^       1339.  Given  two  perpendiculars, 

AB  and  CD,  intersecting  in  0,  and 

a  line  intersecting  these  perpendiculars  in  E  and  F;  construct 
a  square,  one  of  whose  angles  shall  coincide  with  one  of  the  right 
angles  at  0,  and  the  vertex  of  the  opposite  angle  of  the  square 
shall  lie  in  EF.  (Two  solutions.) 

1340.  Draw  from  a  given  point  P  in  the  base  AB  of  a  triangle 
ABC  B,  line  to  AC  produced,  so  that  it  may  be  bisected  by  BC. 

Construct  a  rectangle,  having  given: 

1341.  One  side  and  the  diagonal. 

1342.  One  side  and  the  angle  formed  by  the  diagonals. 

1343.  The  perimeter  and  the  diagonal. 


METHODS  OF  ATTACKING  PROBLEMS          323 

Construct  a  parallelogram,  having  given: 

1344.  Two  independent  sides  and  one  altitude 

1345.  Two  independent  sides  and  an  angle. 

1346.  One  side  and  the  two  diagonals. 

1347.  One  side,  one  angle,  and  one  diagonal. 

1348.  The  diagonals  and  the  angle  formed  by  the  diagonals 
Construct  a  rhombus,  having  given: 

1349.  The  two  diagonals. 

1350.  The  perimeter  and  one  diagonal. 

1351.  One  angle  and  a  diagonal. 

1352.  The  altitude  and  the  base. 

1353.  The  altitude  and  one  angle. 

1354.  Construct  a  square,  having  given  the  diagonal. 
Construct  a  trapezoid,  having  given  : 

1355.  The  four  sides. 

1356.  The  bases,  another  side,  and  one  base  angle. 

1357.  The  bases  and  the  diagonals. 

1358.  One  base,  the  diagonals,  and  the  angle  formed  by  the 
diagonals. 

1359.  Inscribe  a  square  in  a  given  triangle. 

1360.  In  a  given  triangle  inscribe  a  rectangle  similar  to  a  given 
rectangle. 

Hint:  Let  ABC  be  the 
given  triangle  and  R  the 
given  rectangle.  On  the 

altitude  CH    construct  a    A  F         ,B  L 

rectangle  CL  similar  to  the  given  rectangle.     The  line  AK  will  determine  a 
point  E  which  will  be  one  of  the  vertices  of  the  required  rectangle.    Why? 

1361    Inscribe  a  square  in  a  semicircle. 

1362.  Divide  a  given  triangle  into  two  equal  parts  by  a  line 
parallel  to  one  of  the  sides. 

1363.  Bisect  a  triangle  by  a  line  parallel  to  a  given  line. 

1364.  Bisect  a  triangle  by  a  line  drawn  perpendicular  to  the  base. 
Construct  a  circle  which  shall  be  tangent  to  a  given  : 

1365.  Line,  and  to  a  given  circle  at  a  given  point. 

1366.  Circle,  and  to  a  given  line  at  a  given  point. 

1367.  Transform  a  triangle  ABC  so  that  <£A  is  not  altered,  and 


the  side  opposite  the  angle  A  becomes  parallel  to  a  given  line  MN* 


CHAPTER  VIII 

SUMMARIES  AND  APPLICATIONS 

A.  SYLLABUS  OF  THEOREMS 

1.  Vertical  angles  are  equal. 

2.  Two  sides  and  the  included  angle  determine  a  triangle. 

3.  Two  angles  and  the  included  side  determine  a  triangle. 

4.  The  bisector  of  the   vertex  angle  of  an  isosceles  triangle 
divides  it  into  two  congruent  triangles. 

Cor.  1.  The  angles  opposite  the  equal  sides  of  an  isosceles  tri- 
angle are  equal. 

Cor.  2.  The  bisector  of  the  vertex  angle  of  an  isosceles  triangle 
bisects  the  base  and  is  perpendicular  to  it. 

Cor.  3.    An  equilateral  triangle  is  equiangular. 

Cor.  4.  The  bisectors  of  the  angles  of  an  equilateral  triangle 
bisect  the  opposite  sides  and  are  perpendicular  to  them. 

Cor.  5.  The  bisectors  of  the  angles  of  an  equilateral  triangle 
are  equal. 

5.  A  triangle  is  determined  by  its  sides. 

5a.  Only  one  perpendicular  can  be  drawn  through  a  given  point 
to  a  given  line. 

56.  Two  sects  drawn  from  a  point  in  a  perpendicular  to  a  given 
line;  cutting  off  on  the  line  equal  sects  from  the  foot  of  the  per- 
pendicular, are  equal  and  make  equal  angles  with  the  perpendicular. 

5c.  The  sum  of  two  sects  drawn  from  any  point  within  a  triangle  to 
the  ends  of  one  of  its  sides  is  less  than  the  sum  of  its  remaining  sides. 

5d.  Of  two  sects  drawn  from  a  point  in  a  perpendicular  to  a 
given  line  and  cutting  off  unequal  sects  from  the  foot  of  the  per- 
pendicular, the  more  remote  is  the  greater,  and  conversely. 

Cor.  1.  All  possible  obliques  from  a  point  to  a  line  are  equal 
in  pairs,  and  each  pair  cuts  off  equal  sects  from  the  foot 
of  the  perpendicular  from  that  point  to  the  line. 

6.  The  perpendicular  is  the  shortest  sect  from  a  point  to  a  line. 
6a.  The  shortest  sect  from  a  point  to  a  line  is  perpendicular  to  it. 

7.  The  hypotenuse  and  an  adjacent  angle  determine  a  right 
triangle. 

8.  The  hypotenuse  and  another  side  determine  a  right  triangle. 
324 


SUMMARIES  AND  APPLICATIONS  325 

9.  Lines  perpendicular  to  the  same  line  are  parallel. 

10.  A  line  perpendicular  to  one  of  a  series  of  parallels  is  perpen- 
dicular to  the  others 

11.  If  when  lines  are  cut  by  a  transversal  the  alternate-interior 
angles  are  equal,  the  lines  thus  cut  are  parallel. 

Cor.  1.  If  the  alternate-exterior  angles  or  corresponding  angles 
are  equal  when  lines  are  cut  by  a  transversal,  the  lines 
thus  cut  are  parallel. 

Cor.  2.  If  either  the  consecutive-interior  angles  or  the  con- 
secutive-exterior angles  are  supplementary  when  lines 
are  cut  by  a  transversal,  the  lines  thus  cut  are  parallel. 

12.  Parallels  cut  by  a  transversal  form  equal  alternate-interior 
angles. 

Cor.  1.  Parallels  cut  by  a  transversal  form  equal  corresponding 
angles  and  equal  alternate-exterior  angles. 

Cor.  2.  Parallels  cut  by  a  transversal  form  supplementary  con- 
secutive-interior angles  and  supplementary  consecutive 
exterior  angles. 

12a.  Two  angles  whose  sides  are  parallel  each  to  each  or  per- 
pendicular each  to  each,  are  either  equal  or  supplementary. 

13.  The  sum  of  the  angles  of  a  triangle  is  a  straight  angle. 
Cor.  1.    A  triangle  can  have  but  one  right  or  one  obtuse  angle. 
Cor.  2.    Triangles  having  two  angles  mutually  equal  are  mutu- 
ally equiangular. 

Cor.  3.  A  triangle  is  determined  by  a  side  and  any  two  homolo- 
gous angles. 

Cor.  4.  An  exterior  angle  of  a  triangle  is  equal  to  the  sum  of 
the  non-adjacent  interior  angles. 

14.  The  sum  of  the  angles  of  a  polygon  is  equal  to  a  straight 
angle  taken  as  many  times  less  two  as  the  polygon  has  sides. 

Cor.  1.    Each  angle  of  an  equiangular  polygon  of  n  sides  equals 

n 2 

the  th  part  of  a  straight  angle. 

Cor.  2.  The  sum  of  the  exterior  angles  of  a  polygon  is  two 
straight  angles. 

Cor.  3.    Each  exterior  angle  of  an  equiangular  polygon  of  n 

2 
sides  is  equal  to  the  -th  part  of  a  straight  angle. 


326  PLANE  GEOMETRY 

15.  If  two  angles  of  a  triangle  are  equal,  the  sides  opposite  them 
are  equal. 

Cor.  1.    Equiangular  triangles  are  equilateral. 

16.  Either  diagonal  of  a  parallelogram  bisects  it. 

Cor.  1.    The  parallel  sides  of  a  parallelogram  are  equal,  and  the 

opposite  angles  are  equal. 
Cor.  2.    Parallels  are  everywhere  equidistant. 

17.  A  quadrilateral  whose  opposite  sides  are  equal  is  a  parallelo- 
gram. 

18.  A  quadrilateral  having  a  pair  of  sides  both  equal  and  parallel 
is  a  parallelogram. 

19.  A  parallelogram  is  determined  by  two  adjacent  sides  and  an 
angle;  or  parallelograms  are  congruent  if  two  adjacent  sides  and  an 
angle  are  equal  each  to  each. 

20.  The  diagonals  of  a  parallelogram  bisect  each  other. 

21.  A  quadrilateral  whose  diagonals  bisect  each  other  is  a 
parallelogram. 

21a.  The  difference .  between  any  two  sides  of  a  triangle  is  less 
than  the  third  side. 

216.  If  two  sides  of  a  triangle  are  unequal,  the  angles  opposite 
them  are  unequal  in  the  same  order. 

21c.  If  two  angles  in  a  triangle  are  unequal,  the  sides  opposite 
them  are  unequal  in  the  same  order. 

2ld.  If  two  triangles  have  two  sides  equal  each  to  each,  but  the 
included  angles  unequal,  their  third  sides  are  unequal  in  the  same 
order  as  those  angles. 

2le.  If  two  triangles  have  two  sides  equal  each  to  each,  but  the 
third  sides  unequal,  the  angles  opposite  those  sides  are  unequal 
in  the  same  order. 

2 1/.  If  one  acute  angle  of  a  right  triangle  is  double  the  other 
the  hypotenuse  is  double  the  shorter  leg,  and  conversely. 

22.  Rectangles  having  a  dimension  of  one  equal  to  that  of 
another  compare  as  their  remaining  dimensions. 

23.  Any   two   rectangles    compare   as   the   products   of   their 
dimensions. 

24.  The  area  of  a  rectangle  is  equal  to  the  product  of  its  base 
and  altitude. 


SUMMARIES  AND  APPLICATIONS  327 

25.  The  area  of  a  parallelogram  is  equal  to  the  product  of  its 
base  and  altitude. 

Cor.  1.    Any  two  parallelograms  compare  as  the  products  of 

their  bases  and  altitudes. 
Cor.  2.    Parallelograms  having  one  dimension  equal  compare  as 

their  remaining  dimensions. 
Cor.  3.    Parallelograms  having  equal  bases  and  equal  altitudes 

are  equal. 

26.  The  area  of  a  triangle  is  equal  to  half  the  product  of  its 
base  and  altitude. 

Cor.  1.    Any  two  triangles  compare  aa  the  products  of  their 

bases  and  altitudes. 
Cor.  2.    Triangles  having  one  dimension  equal  compare  as  their 

remaining  dimensions. 
Cor.  3.    Triangles  having  equal  bases  and  equal  altitudes  are 

equal. 

26a.  The  square  on  the  hypotenuse  of  a  right  triangle  equals 
the  sum  of  the  squares  on  the  two  legs. 

266.  The  areas  of  two  triangles  that  have  an  angle  of  one  equal 
to  an  angle  of  the  other,  are  to  each  other  as  the  products  of  the 
sides  including  those  angles. 

27.  The  area  of  a  trapezoid  is  equal  to  half  the  product  of  its 
altitude  and  the  sum  of  its  bases. 

28.  Any  proportion  may  be  transformed  by  alternation,  i.e.,  the 
first  term  is  to  the  third  as  the  second  is  to  the  fourth. 

29.  In  any  proportion  the  terms  may  be  combined  by  addition 
(usually  called  composition) ;  i.e.,  the  ratio  of  the  sum  of  the  first 
and  second  terms  to  the  second  term  (or  first  term)  equals  the  ratio 
of  the  sum  of  the  third  and  fourth  terms  to  the  fourth  term  (or 
the  third  term). 

N.B. — "Addition"  and  "sum"  are  used  in  the  algebraic  sense. 

30.  In  a  series  of  equal  ratios,  the  ratio  of  the  sum  of  any  num- 
ber of  antecedents  to  the  sum  of  their  consequents  equals  the 
ratio  of  any  antecedent  to  its  consequent. 

31.  A  line  parallel  to  one  side  of  a  triangle  divides  the  other 
sides  proportionally. 


328  PLANE  GEOMETRY 

Cor.  1.  One  side  of  a  triangle  is  to  either  of  the  sects  cut  off 
by  a  line  parallel  to  a  second  side  as  the  third  side  is 
to  its  homologous  sect. 

Cor.  2.    Parallels  cut  off  proportional  sects  on  all  transversals. 

Cor.  3.  Parallels  which  intercept  equal  sects  on  one  trans- 
versal do  so  on  all  transversals. 

Cor.  4.  A  line  which  bisects  one  side  of  a  triangle,  and  is  parallel 
to  the  second,  bisects  the  third. 

Cor.  5.  A  sect  which  bisects  two  sides  of  a  triangle  is  parallel 
to  the  third  side  and  equal  to  half  of  it. 

Cor.  6.  The  line  (usually  called  median)  joining  the  mid-points 
of  the  non-parallel  sides  of  a  trapezoid  is  parallel  to  the 
bases  and  equal  to  one-half  their  sum. 

Cor.  7.  The  area  of  a  trapezoid  equals  the  product  of  its  median 
and  altitude. 

32.  A  line  dividing  two  sides  of  a  triangle  proportionally  is 
parallel  to  the  third  side. 

Cor.  1.  A  line  dividing  two  sides  of  a  triangle  so  that  these 
sides  bear  the  same  ratio  to  a  pair  of  homologous  sects, 
is  parallel  to  the  third  side. 

32a.  The  bisector  of  an  angle  of  a  triangle  divides  the  opposite 
side  into  sects  which  are  proportional  to  the  adjacent  sides. 

326.  The  bisector  of  an  exterior  angle  of  a  triangle  divides  the 
opposite  side  externally  into  sects  which  are  proportional  to  the 
adjacent  sides. 

Cor.  1.  The  bisectors  of  an  adjacent  interior  and  exterior  angle 
of  a  triangle  divide  the  opposite  side  harmonically. 

33.  The  homologous  angles  of  similar  triangles  are  equal  and 
their  homologous  sides  have  a  constant  ratio. 

34.  Triangles  are  similar  when  two  angles  of  one  are  equal  each 
to  each  to  two  angles  of  another. 

34a.  Triangles  which  have  their  sides  parallel  or  perpendicular 
each  to  each  are  similar. 

35.  Triangles  which  have  two  sides  of  one  proportional  to  two 
sides  of  another,  and  the  included  angles  equal,  are  similar. 

36.  If  the  ratio  of  the  sides  of  one  triangle  to  those  of  another 
is  constant,  the  triangles  are  similar. 


SUMMARIES  AND  APPLICATIONS          -      329 

36a.  The  homologous  angles  of  similar  polygons  are  equal,  and 
their  homologous  sides  have  a  constant  ratio. 
Cor.  1.    The  homologous  diagonals  drawn  from  a  single  vertex 

of  similar  polygons  divide  the  polygons  into  triangles 

similar  each  to  each. 

366.  Polygons  are  similar  if  their  homologous  angles  are  equal 
and  homologous  sides  proportional. 

Cor.  1.    If  homologous  diagonals  drawn  from  a  single  vertex  of 

polygons  divide  them  into  triangles  similar  each  to  each, 

the  polygons  are  similar. 

37.  The  perimeters  of  similar  triangles  are  proportional  to  any 
two  homologous  sides,  or  any  two  homologous  altitudes. 

Cor.  1.    Homologous  altitudes  of  similar  triangles  have  the  same 

ratio  as  homologous  sides. 
Cor.  2.    The  perimeters  of  similar  polygons  have  the  same  ratio 

as  any  pair  of  homologous  sides  or  diagonals. 

38.  The  areas  of  similar  triangles  compare  as  the  squares  of  any 
two  homologous  sides. 

Cor.  1.    The  areas  of  similar  polygons  compare  as  the  squares 

of  any  two  homologous  sides  or  diagonals. 
Cor.  2.    Homologous  sides  or  diagonals  of  similar  polygons  have 

the  same  ratio  as  the  square  roots  of  the  areas. 
38a.  If  two  parallels  are  cut  by  concurrent  transversals,  the  ratio 
of  homologous  sects  of  the  parallels  is  constant. 

386.  If  the  ratio  of  homologous  sects  of  two  parallels  cut  by 
three  or  more  transversals  is  constant,  the  transversals  are  either 
parallel  or  concurrent. 

39.  The  altitude  upon  the  hypotenuse  of  a  right  triangle  divides 
it  into  triangles  similar  to  each  other  and  to  the  original. 

Cor.  1.    Each  side  of  a  right  triangle  is  a  mean  proportional 

between  the  hypotenuse  and  its  projection  upon  the 

hypotenuse. 
Cor.  2.    The  square  of  the  hypotenuse  of  a  right  triangle  is 

equal  to  the  sum  of  the  squares  of  the  other  two  sides. 
Cor.  3.    The  altitude  upon  the  hypotenuse  of  a  right  triangle 

is  a  mean  proportional  between  the  sects  it  cuts  off  on 

the  hypotenuse. 


330  PLANE  GEOMETRY 

39a.  In  any  triangle,  the  square  of  the  side  opposite  an  acute 
angle  is  equal  to  the  sum  of  the  squares  of  the  other  two  sides, 
diminished  by  twice  the  product  of  one  of  those  sides  by  the  pro- 
jection of  the  other  upon  it. 

396.  In  an  obtuse  triangle,  the  square  of  the  side  opposite  the 
obtuse  angle  is  equal  to  the  sum  of  the  squares  of  the  other  two 
sides,  increased  by  twice  the  product  of  one  of  those  sides  by  the 
projection  of  the  other  upon  it. 

39c.  I.  The  sum  of  the  squares  of  two  sides  of  a  triangle  is  equal 
to  twice  the  square  of  half  the  third  side,  increased  by  twice  the 
square  of  the  median  to  it. 

II.  The  difference  of  the  squares  of  two  sides  of  a  triangle  is 
equal  to  twice  the  product  of  the  third  side  and  the  projection  of 
the  median  upon  it. 

Cor.  1.  If  ma  represents  the  length  of  the  median  upon  side  a 
of  the  triangle  whose  sides  are  a,  6,  c,  then  wa= J\/2(62+c2)  —a2. 

39d.  If  h  a  represents  the  altitude  upon  side  a  of  a  triangle  whose 
sides  are  a,  6,  c,  and  s  stands  for  its  semi  perimeter,  i.e., 

s=  —       — ,  then  ha=-\/s(s  -a)(s  —  b)(s  —  c). 
Zi  a 

Cor.  1.  If  A  stands  for  the  area  of  a  triangle  whose  sides  are 
a,  6,  c,  and  whose  semiperimeter  is  s,  then  A  =  \/s(s — a)  (s  —b)  (s—c) . 

39e.  If  similar  polygons  are  constructed  on  the  sides  of  a  right 
triangle,  as  homologous  sides,  the  polygon  on  the  hypotenuse  is 
equal  to  the  sum  of  the  polygons  on  the  other  two  sides. 

40.  The  locus  of  points  equidistant  from  the  ends  of  a  sect  is 
the  perpendicular  bisector  of  the  sect. 

Cor.  1.  Two  points  equidistant  from  the  ends  of  a  sect  fix  its 
perpendicular  bisector. 

41.  The  locus  of  points  equidistant  from  the  sides  of  an  angle 
is  the  bisector  of  the  angle. 

Cor.  1.  The  locus  of  a  point  equidistant  from  two  intersecting 
lines  is  a  pair  of  lines  bisecting  the  angles. 

41a.  The  bisectors  of  the  angles  of  a  triangle  are  concurrent  in  a 
point  equidistant  from  the  sides  of  the  triangle. 

416.  The  perpendicular  bisectors  of  the  sides  of  a  triangle  are 
concurrent  in  a  point  equidistant  from  the  vertices. 

41c.  The  altitudes  of  a  triangle  are  concurrent. 


SUMMARIES  AND  APPLICATIONS  331 

The  medians  of  a  triangle  are  concurrent  in  a  point  of  tri- 
section  of  each. 

42.  Three  points  not  in  a  straight  line  fix  a  circle. 

43.  In  equal  circles,  equal  central  angles  intercept  equal  arcs, 
and  conversely. 

43a.  In  equal  circles,  the  greater  of  two  central  angles  intercepts 
the  greater  arc,  and  conversely. 

44.  In  equal  circles,  equal  arcs  are  subtended  by  equal  chords, 
and  conversely. 

44a.  In  equal  circles,  unequal  arcs  are  subtended  by  chords 
unequal  in  the  same  order,  and  conversely. 

45.  A  diameter  perpendicular  to  a  chord  bisects  it  and  its  sub- 
tended arcs. 

Cor.  1.    A  radius  which  bisects  a  chord  is  perpendicular  to  it. 
Cor.  2.    The  perpendicular  bisector  of  a  chord  passes  through 
the  center  of  the  circle. 

46.  In  equal  circles,  equal  chords  are  equidistant  from  the  center, 
and  conversely. 

46a.  In  equal  circles,  the  distances  of  unequal  chords  from  the 
center  are  unequal  in  the  opposite  order. 

47.  A  line  perpendicular  to  a  radius  at  its  outer  extremity  is 
tangent  to  the  circle. 

Cor.  1.    A  tangent  to  a  circle  is  perpendicular  to  the  radius 

drawn  to  the  point  of  contact. 
Cor.  2.    The  perpendicular  to  a  tangent  at  the  point  of  contact 

passes  through  the  center  of  the  circle. 
Cor.  3.    A  radius  perpendicular  to  a  tangent  passes  through  the 

point  of  contact. 
Cor.  4.    Only  one  tangent  can  be  drawn  to  a  circle  at  a  given 

point  on  the  circle. 

48.  Sects  of  tangents  from  the  same  point  to  a  circle  are  equal. 

49.  The  line  of  centers  of  two  tangent  circles  passes  through 
their  point  of  contact. 

49a.  The  line  of  centers  of  two  intersecting  circles  is  the  perpen- 
dicular bisector  of  their  common  chord. 

50.  In  equal  circles  central  angles  have  the  same  ratio  as  their 
intercepted  arcs. 

Cor.  1.    A  central  angle  is  measured  by  its  intercepted  arc. 


332  PLANE  GEOMETRY 

51.  Parallels  intercept  equal  arcs  on  a  circle. 

52.  An  inscribed  angle,  or  one  formed  by  a  tangent  and  a  chord 
is  measured  by  one-half  its  intercepted  arc. 

52a.  The  mid-point  of  the  hypotenuse  of  a  right  triangle  is 
equidistant  from  the  three  vertices. 

53.  An  angle  whose  vertex  is  inside  the  circle  is  measured  by 
half  the  sum  of  the  arcs  intercepted  by  it  and  its  vertical. 

54.  An  angle  whose  vertex  is  outside  the  circle  is  measured  by 
half  the  difference  of  its  intercepted  arcs. 

54a.  The  opposite  angles  of  a  quadrilateral  inscribed  in  a  circle 
are  supplementary. 

Cor.  1.  A  quadrilateral  is  inscriptible  if  its  opposite  angles  are 
supplementary. 

55.  A  tangent  is  the  mean  proportional  between  any  secant  and 
its  external  sect,  when  drawn  from  the  same  point  to  a  circle. 

Cor.  1.  The  product  of  a  secant  and  its  external  sect  from  a 
fixed  point  outside  a  circle  is  constant. 

55a.  If  chords  intersect  inside  a  circle,  the  product  of  their  sects 
is  constant. 

556.  The  square  of  the  bisector  of  an  angle  of  a  triangle  is  equal 
to  the  product  of  the  sides  of  this  angle,  diminished  by  the  product 
of  the  sects  made  by  the  bisector  on  the  third  side. 

55c.  In  any  triangle  the  product  of  two  sides  is  equal  to  the 
product  of  the  diameter  of  the  circumscribed  circle  and  the  alti- 
tude on  the  third  side. 

Cor.  1.  If  R  denote  the  radius  of  the  circle  circumscribed  about 
a  triangle  whose  sides  are  a,  6,  c,  and  semiperimeter  s, 

then  R=-  -=       abc        _,. 


56.  A  circle  may  be  circumscribed  about,  and  inscribed  within 
any  regular  polygon. 

Cor.  1.    An  equilateral  polygon  inscribed  in  a  circle  is  regular. 
Cor.  2.    An  equiangular  polygon  circumscribed  about  a  circle  is 

regular. 
Cor.  3.    The  area  of  a  regular  polygon  is  equal  to  half  the  product 

of  its  apothem  and  perimeter. 


SUMMARIES  AND  APPLICATIONS  333 

57.  If  a  circle  is  divided  into  any  number  of  equal  arcs,  the  chords 
joining  the  successive  points  of  division  form  a  regular  inscribed 
polygon;  and  the  tangents  drawn  at  the  points  of  division  form  a 
regular  circumscribed  polygon. 

Cor.  1.  Tangents  to  a  circle  at  the  vertices  of  a  regular  inscribed 
polygon  form  a  regular  circumscribed  polygon  of  the 
same  number  of  sides. 

Cor.  2  Lines  drawn  from  each  vertex  of  a  regular  inscribed 
polygon  to  the  mid-points  of  the  adjacent  arcs  sub- 
tended by  its  sides,  form  a  regular  inscribed  polygon  of 
double  the  number  of  sides. 

Cor.  3.  Tangents  at  the  mid-points  of  the  arcs  between  conse- 
cutive points  of  contact  of  the  sides  of  a  regular  circum- 
scribed polygon  form  a  regular  circumscribed  polygon 
of  double  the  number  of  sides. 

Cor.  4.  The  perimeter  of  a  regular  inscribed  polygon  is  less 
than  that  of  a  regular  inscribed  polygon  of  double  the 
number  of  sides,  and  the  perimeter  of  a  regular  circum- 
scribed polygon  is  greater  than  that  of  a  regular  cir- 
cumscribed polygon  of  double  the  number  of  sides. 

Cor.  5.  Tangents  to  a  circle  at  the  mid-points  of  the  arcs  sub- 
tended by  the  sides  of  a  regular  inscribed  polygon 
form  a  regular  circumscribed  polygon,  of  which  the 
sides  are  parallel  to  the  sides  of  the  original  and  the 
vertices  lie  on  the  prolongations  of  the  radii  of  the 
inscribed  polygon. 

58.  A  regular  polygon  the  number  of  whose  sides  is  3-2n  may  be 
inscribed  in  a  circle. 

59.  If  in  represent  the  side  of  a  regular  inscribed  polygon  of 
n  sides,  and  iZn  the  side  of  one  of  2n  sides,  and  r  the  radius  of  the 

circle,  i2n  =  A/2r2  -rA/4r2  -in*. 

60.  If  in  represent  the  side  of  a  regular  inscribed  polygon  of  n 
sides,  cn  that  of  a  regular  circumscribed  polygon  of  n  sides,  and  r 

the  radius  of  the  circle,  cn=  n 

V  4r2  -  inz 

61.  The  perimeters  of  regular  polygons  of  the  same  number  of 
sides  compare  as  their  radii  and  also  as  their  apothems. 


334  PLANE  GEOMETRY 

62.  Circumferences  have  the  same  ratio  as  their  radii. 

Cor.  1.    The  ratio  of  any  circle  to  its  diameter  is  constant. 
Cor.  2.    In  any  circle  c  =  2-nr. 

63.  The  value  of  TT  is  approximately  3.14159. 

64.  The  area  of  a  circle  is  equal  to  one-half  the  product  of  its 
radius  and  its  circumference. 

Cor.  1.    The  area  of  a  circle  is  equal  to  TT  times  the  square 

of  its  radius. 
Cor.  2.    The  areas  of  circles  compare  as  the  squares  of  their 

radii. 
Cor.  3.    The  area  of  a  sector  is  equal  to  half  the  product  of 

its  radius  and  its  arc. 
Cor.  4.    Similar  sectors  and  similar  segments  compare  as  the 

squares  of  their  radii. 

B.  SYLLABUS  OF  CONSTRUCTIONS 

Problem  1.  Draw  a  perpendicular  to  a  given  line  from  a  given 
point  (a)  outside  the  line,  (6)  on  the  line. 

Problem  2.    Bisect  a  given  (a)  sect,  (6)  angle,  (c)  arc. 

Problem  3.    Reproduce  a  given  angle. 

Problem  4.  Draw  a  line  through  a  given  point,  and  parallel  to 
,a  given  line. 

Problem  6.  Construct  a  triangle  given  any  three  independent 
parts;  (a)  two  angles  and  the  included  side,  (6)  two  sides  and  the 
included  angle,  (c)  three  sides,  (d)  the  hypotenuse  and  a  leg  of  a 
right  triangle. 

Problem  6.    Divide  a  sect  into  n  equal  parts. 

Problem  7.  Find  a  common  measure  of  two  commensurable 
sects. 

Problem  8.    Pass  a  circle  through  three  non-collinear  points. 
Cor.  1.    Circumscribe  a  circle  about  a  triangle. 

Problem  9.  Divide  a  given  sect  into  parts  proportional  to  n 
given  sects. 

Problem  10.  Divide  a  sect  harmonically  in  the  ratio  of  two 
given  sects. 

Problem  11.    Find  a  fourth  proportional  to  three  given  sects. 
Cor.  1.    Find  a  third  proportional  to  two  given  sects. 

Problem  12.    Upon  a  given  sect  as  homologous  to  a  designated 


SUMMARIES  AND  APPLICATIONS  335 

side  of  a  given  polygon  construct  another  similar  to  the  original 
polygon. 

Problem  13.  Construct  a  square  equal  to  the  sum  of  two  or 
more  given  squares. 

Cor.  1.    Construct  a  square  equal  to  the  difference  of  two 

given  squares. 

Cor.  2.    Construct  a  polygon  similar  to  two  given  similar  poly- 
gons and  equal  to  (a)  their  sum,  and  (6)  their  difference. 
Problem  14.    Inscribe  in  a  circle  regular  polygons  the  number 
of  whose  sides  is  (a)  3-2n,  (6)  4-2n. 

Problem  15.  Construct  a  triangle,  given  two  sides  and  the  angle 
opposite  one  of  them. 

Problem  16.  Through  a  given  point  draw  a  tangent  to  a  given 
circle  (a)  when  the  point  is  on  the  circle,  (6)  when  the  point  is 
outside  the  circle. 

Problem  17.    Construct  a  common  (a)  external  and  (6)  internal 
tangent  to  two  circles. 
Problem  18.     Inscribe  a  circle  in  a  triangle. 

Cor.  1.    Find  the  centers  of  the  escribed  circles  of  a  triangle. 
Problem  19.    Construct  upon  a  given  sect  the  segment  of  a 
circle  capable  of  containing  a  given  angle. 

Problem  20.  Construct  a  mean  proportional  between  two  given 
sects. 

Problem  21.    Divide  a  given  sect  into  mean  and  extreme  ratio. 
Problem  22.    Inscribe  in  a  circle  a  regular  decagon. 

Cor.  1.    Inscribe  in  a  circle  regular  polygons  the  number  of 

whose  sides  is  (a)  5-2n,  (6)  15-2n. 
Problem  23.    Transform  a  polygon  into  a  triangle. 
Problem  24.    Construct  the  square  equal  to  a  given  (a)  parallelo- 
gram, (6)  triangle,  (c)  polygon. 

Problem  25.  Construct  a  parallelogram  equal  to  a  given  square, 
having  given  (a)  the  sum  of  base  and  altitude  equal  to  a  given 
sect,  (6)  the  difference  of  base  and  altitude  equal  to  a  given  sect. 

Problem  26.  Construct  a  polygon  similar  to  one  and  equal  to 
another  of  two  given  polygons. 

Problem  27.  Construct  a  square  which  shall  have  a  given  ratio 
to  a  given  square. 

Cor.  1.    Construct  a  polygon  equal  to  any  part  of  a  given 
polygon  and  similar  to  it. 


336  PLANE  GEOMETRY 

C.  SUMMARY  OF  FORMULAS 

I.  SUM  OF  THE  ANGLES  OF  A  POLYGON. 

a.  Interior.  S=n—  2.    (S  stands  for  sum  of  angles  in 

~,     .  ,.,    n  —  2     (E  stands  for  each  angle  of  an 

Equiangular.    E= .  ,  v  , 

n  equiangular  n-gon,  in  st.  <£«.) 

6.  Exterior.  £=2. 

2 

Equiangular.    E=—. 

II.  AREAS. 

a.  Parallelogram.     A=bh. 

b.  Triangle  A=%bh. 

c.  Trapezoid.       1.  A=%h(b+bi). 

2.  A=hm. 

in.  METRIC  RELATIONS  IN  TRIANGLES. 

o.  Right  triangle.     C2=a2-f62.    (^CC=H.<JC.) 
6.  Acute  triangle.    c2=a2+62  -2apba. 

c.  Obtuse  triangle.  cz=az+bz+2apba. 

d.  In  any  triangle. 


2.  a2  -bz=2cp 


3.  mc=-h/2(a2+fr2)  -c2. 


5.  ab=hcD.    (D  stands  for  diameter  of  circum- 

scribed circle.) 

6.  te2=ab-pq.  (p  and  q  are  sects  of  c  made  by  t.) 


8.  A  =  \/s(s-a)(s-b)(s-c). 
IV.  MENSURATION  OF  THE  CIRCLE. 

a.  12  =  —          abc       d    C=2irr  or  C=irD. 

4  v  s(s  -a)  (s  -6)  (s  -c) 

b.  i'2n=V2r2-rV4r2-i«2.  e.  ^=3.14159.... 

^^7rr2orA=-- 


SUMMARIES  AND  APPLICATIONS  337 


V.  RATIO  AND  PROPORTION. 

Given:   |-  =  -jfthen, 

a.  Product  of  means  ----      ad=bc. 

6.  Alternation.  —  a  -=-. 

c       d 

b      d 

c.  Inversion.  —  =  —  . 

a       c 


d.  Addition  (Composition)          _  m  -       or       -  ^ 

o  d  a  c 


e.  Series  of  equal  ratios.     Given   ^  s  ^  =  ~  B  .  .  .,  then, 


VI.  DIVISION  OF  SECTS. 


TT  .      A/     AE  A 

a.  Harmonu;. 


6.  Extreme  and  mean  ratio. 


AI  ~IB  a  d  BE~AE 

VII.  SIMILAR  FIGURES. 

P=£     (P  and  p  stand  for  perimeter,  S  and  s  for 
p     s '        homologous  sides.) 

6.  -=-*. 

r-I  A    JfE2 

c.  Circle.    1,  -  =  — -. 

a       r2 

c      r      'd,' 
22 


338  PLANE  GEOMETRY 

VIH.  MEASUREMENT  OF  ANGLES. 

o.    Coca.    (C  stands  for  number  of  degrees  in  central  angle.) 

6.     /°£~.          (/  stands  for  number  of  degrees  in  inscribed  angle 
or  one  formed  by  tangent  and  chord.) 


c.  Tfoc.     (W  stands  for  number  of  degrees  in  angle  with 

vertex  inside  circle.) 

I  —s 

d.  Ooc  —  .     (0  stands  for  number  of  degrees  in  angle  with 

vertex  outside  circle.) 

(o  stands  for  number  of  degrees  in  intercepted  arc.) 
(v  stands  for  number  of  degrees  in  arc  intercepted  by  vertical 
angle.) 

(I  stands  for  number  of  degrees  in  larger  intercepted  arc.) 
(s  stands  for  number  of  degrees  in  smaller  intercepted  arc.) 

IX.  TANGENT  AND  SECANT. 

a.  T*=S-E.    (T  stands  for  length  of  sect  of  tangent  from  point 

to  circle.) 
(S  stands  for  length  of  secant  from  same  point 

to  circle.) 

(E  stands  for  length  of  external  sect  of  the  secant.) 
6.  pq=ab.       (pq  stand  for  sects  of  a  chord  made  by  the  inter- 
section of  a  chord  whose  sects  are  a  and  6.) 

D.  SUMMARY  OF  METHODS  OF  PROOF 
I.  TRIANGLES  CONGRUENT. 
Show  that  they  have 
(a)  two  sides  and  included  angle  respectively  equal, 
(6)  a  side  and  any  two  angles  respectively  equal, 

(c)  three  sides  respectively  equal, 
or,  that  they  are  parts  of 

(d)  an  isosceles  triangle  formed  by  the  bisector  of  the  vertex 

angle, 

(e)  a  parallelogram  formed  by  the  diagonal, 
or,  that  they  are  right  triangles,  and  have 

(/)   the  hypotenuse  and  adjacent  angle  respectively  equal, 
(g)  the  hypotenuse  and  adjacent  side  respectively  equal. 


SUMMARIES  AND  APPLICATIONS  339 

EXERCISES.    SET  XCVI.    CONGRUENCE  OF  TRIANGLES 

1368.  If  a  perpendicular  is  erected  at  any  point  on  the  bisector 
cf  an  angle  and  produced  to  cut  the  sides  of  the  angle,  two  con- 
gruent triangles  are  formed. 

1369.  In  triangle  ABC,  BD  (D  in  AC),  the  bisector  of  angle  B 
intersects  AE  (E  in  BC),  a  perpendicular  to  BD,  in  F.    Prove  that 
triangles  ABF  and  BFE  are  congruent. 

1370.  Triangles  are  congruent  if  two  sides  and  the  median  to 
one  of  these  sides  are  equal  respectively  to  two  sides  and  the  homol- 
ogous median  in  the  other.     (Is  this  always  true  or  is  there  any 
exception?) 

1371.  Isosceles  triangles  are  congruent  if  the  base  and  a  leg  of 
one  are  equal  respectively  to  the  base  and  a  leg  of  the  other. 

1372.  The  three  sects  joining  the  mid-points  of  the  sides  of  a 
triangle  divide  the  figure  into  four  congruent  triangles. 

1373.  If  equal  distances  are  laid  off  from  the  same  vertex  on 
the  legs  of  an  isosceles  triangle,  and  these  points  of  division  are 
joined  with  the  opposite  vertices,  congruent  triangles  are  formed. 

1374.  If  two  angles  of  a  triangle  are  equal,  the  bisector  of  the 
third  angle  divides  the  figure  into  two  congruent  triangles. 

II.  SECTS  EQUAL. 
Show  that  they  are 

(a)  homologous  parts  of  congruent  polygons, 
(6)  sects  from  a  point  in  a  perpendicular  cutting  off  equal 
distances  from  its  foot, 

(c)  sects  of  a  line  cut  off  from  the  foot  of  a  perpendicular  to 

it  by  equal  sects  from  the  same  point  in  the  perpen- 
dicular, 

(d)  bisectors  of  the  angles  of  an  equilateral  triangle, 

(e)  sects  of  the  base  of  an  isosceles  triangle  formed  by  the 

bisector  of  the  vertex  angle, 
(/)   sides  opposite  equal  angles  of  a  triangle, 
(g)  parallel  sides  of  a  parallelogram, 

(h)  sects  of  one  diagonal  of  a  parallelogram  made  by  the  other, 
(i)   sects  on  a  transversal  cut  off  by  parallels  which  cut  off 

equal  sects  on  some  other  transversal, 


340  PLANE  GEOMETRY 

(j)   chords  in  equal  circles  subtending  equal  arcs, 

(k)  chords  in  equal  circles  equally  distant  from  the  center, 

(I)   sects  of  tangents  to  a  circle  from  the  same  point. 

EXERCISES.    SET  XCVII.    EQUALITY  OF  SECTS 

1375.  The  bisectors  of  the  base  angles  of  an  isosceles  triangle 
are  equal. 

1376.  The  sects  joining  the  mid-points  of  the  legs  of  an  isosceles 
triangle  with  the  mid-point  of  the  base  are  equal. 

1377.  The  medians  to  the  legs  of  an  isosceles  triangle  are  equal. 

1378.  If  the  base  of  an  isosceles  triangle  is  trisected,  the  sects 
joining  the  points  of  division  with  the  vertex  are  equal. 

1379.  If  from  any  point  in  the  circumference  of  a  circle  two 
chords  are  drawn  making  equal  angles  with  the  radius  to  the  point, 
these  chords  are  equal. 

1380.  A  circumscribed  parallelogram  is  equilateral. 

HI.  ANGLES  EQUAL. 
Show  that  they  are 

(a)  straight  or  right  angles, 

(6)  supplements  or  complements  of  equal  angles, 

(c)  vertical  angles, 

(d)  homologous  parts  of  congruent  polygons, 

(e)  angles  opposite  the  equal  sides  of  a  triangle, 

(/)   alternate-interior,    alternate-exterior,    or    corresponding 

angles  of  parallels, 

(g)  opposite  angles  of  a  parallelogram, 
(h)  homologous  angles  of  similar  polygons, 
(i)   measured  by  equal  arcs, 
or,  that  their  sides  are  respectively 
(j)   perpendicular  or  parallel. 

EXERCISES.    SET  XCVIII.    EQUALITY  OF  ANGLES 

1381.  The  tangent  at  the  vertex  of  an  inscribed  equilateral 
triangle  forms  equal  angles  with  the  adjacent  sides. 

1382.  If  in  the  base  AC  of  an  isosceles  triangle  ABC,  two  points 
D  and  E  are  taken  so  that  AE=AB  and  CD  =  BC,  prove  that 


SUMMARIES  AND  APPLICATIONS  341 

1383.  If  triangles  are  similar  to  the  same  triangle,  they  are 
similar  to  each  other. 

1384.  If  through  the  vertices  of  an  isosceles  triangle  lines  are 
drawn  parallel  to  the  opposite  sides,  they  form  an  isosceles  triangle. 

1385.  If  the  bisector  of  an  exterior  angle  of  a  triangle  is  parallel 
to  the  opposite  side,  the  triangle  is  isosceles. 

1386.  Two  circles  intersect  at  the  points  A  and  B.    Through 
A  a  variable  secant  is  drawn,  cutting  the 

circles  at  Cand  D.    Prove  that  the  angle 
DEC  is  constant. 

1387.  Two  circles  touch  each  other  inter- 
nally at  P.    MN  is  a  chord  of  the  larger, 
tangent  to  the  smaller  at  C.    Prove 


IV.  LINES  PARALLEL. 
Show  that  they  are 

(a)  perpendicular  or  parallel  to  the  same  line, 
(6)  opposite  sides  of  a  quadrilateral  which  can  be  shown  to 
be  a  parallelogram, 

(c)  one  side  of  a  triangle  and  a  sect  cutting  the  other  two 

sides  proportionally, 

(d)  side  of  a  regular  inscribed  polygon  and  the  side  of  a  polygon 

the  sides  of  which  are  tangent  to  the  circle  at  the  mid- 
points of  the  arcs  subtended  by  the  inscribed  polygon, 
or,  that  they  have 

(e)  alternate-interior,    alternate-exterior,    or    corresponding 

angles  equal, 

(/)   consecutive-interior  or  consecutive-exterior  angles  supple- 
mentary. 

EXERCISES.    SET  XCIX.    PARALLELISM  OF  LINES 

1388.  If  two  sides  of  a  triangle  are  produced  their  own  lengths 
through  the  common  vertex,  a  line  joining  their  ends  is  parallel 
to  the  third  side  of  the  triangle. 

1389.  The  line  joining  the  feet  of  the  perpendiculars  dropped 
from  the  extremities  of  the  base  of  an  isosceles  triangle  to  the 
opposite  sides  is  parallel  to  the  base. 


342  PLANE  GEOMETRY 

1390.  The  tangents  drawn  through  the  extremities  of  a  diameter 
are  parallel. 

1391.  If  from  one  pair  of  opposite  vertices  of  a  parallelogram 
lines  are  drawn  bisecting  the  opposite  sides  respectively,  the  lines 
are  parallel. 

1392.  The  bisectors  of  a  pair  of  opposite  angles  of  a  parallelo- 
gram are  parallel. 

1393.  If  two  circles  intersect  and  a  sect  be  drawn  through  each 
point  of  intersection  terminated  by  the  circumferences,  the  chords 
which  join  the  extremities  of  these  sects  are  parallel. 

1394.  If  through  the  point  of  contact  of  two  tangent  circles  two 
secants  are  drawn,  the  chords  joining  the  points  where  the  secants 
cut  the  circles  are  parallel.    Discuss  both  cases. 

1395.  If  a  straight  line  be  drawn  through  the  point  of  contact 
of  two  tangent  circles  so  as  to  form  chords,  the  radii  drawn  to  the 
other  extremities  of  these  chords  are  parallel.    (What  two  cases?) 

1396.  If  a  straight  line  be  drawn  through  the  point  of  contact 
of  two  tangent  circles  so  as  to  form  chords,  the  tangents  drawn  at 
the  other  extremities  of  the  chords  are  parallel.    (What  two  cases?) 
V.  LINES  PERPENDICULAR. 

Show  that 

(a)  two  adjacent  angles  formed  by  them  are  equal, 
(6) .  one  is  the  base  and  the  other  the  bisector  of  the  vertex 
angle  of  an  isosceles  triangle, 

(c)  one  of  them  is  perpendicular  to  a  line  parallel  to  the  other, 

(d)  one  is  a  tangent  to  a  circle  and  the  other  a  radius  drawn 

to  the  point  of  contact, 

(e)  one  is  a  radius  bisecting  the  other  which  is  a  chord  of  the 

same  circle, 

(/)   one  is  the  shortest  sect  from  a  point  to  the  other. 
EXERCISES.    SET  C.    PERPENDICULARITY  OF  LINES 

1397.  Every  parallelogram  inscribed  in  a  circle  is  a  rectangle. 

1398.  The  bisectors  of  two  interior  angles  on  the  same  side  of  a 
transversal  to  two  parallels  are  perpendicular  to  each  other. 

1399.  If  either  leg  of  an  isosceles  triangle  be  produced  through 
the  vertex  by  its  own  length,  and  the  extremity  joined  to  the 
extremity  of  the  base,  the  joining  line  is  perpendicular  to  the  base« 


SUMMARIES  AND  APPLICATIONS  343 

1400.  Two  circles  are  tangent  externally  at  A,  and  a  common 
external  tangent  touches  them  in  B  and  C.     Show  that  angle 
BAG  is  a  right  angle. 

1401.  If  two  consecutive  angles  of  a  quadrilateral  are  right 
angles,  the  bisectors  of  the  other  two  angles  are  perpendicular. 

1402.  The  line  joining  the  center  of  a  circle  to  the 
mid-point  of  a  chord  is  perpendicular  to  the  chord. 

1403.  If  the  diagonals  of  a  parallelogram  are  equal 
the  figure  is  a  rectangle. 

1404.  If  the  opposite  sides  of 

an  inscribed  quadrilateral  be  produced  to  meet 
in  A  and  F,  the  bisectors  of  the  angles  A  and  F 
meet  at  right  angles. 

Hint:  Prove  %.BGK=%.CHK 

VI.  SECTS  UNEQUAL. 

Show  that  they  are 

(a)  sects  from  a  point  in  a  perpendicular  to  a  line  cutting  off 

unequal  sects  from  its  foot, 
(6)  sects  on  a  line  cut  off  by  unequal  sects  drawn  from  a 

point  in  the  perpendicular  to  that  line, 

(c)  sides  of  a  triangle  opposite  unequal  angles, 

(d)  third  sides  of  two  triangles  which  have  the  other  two  sides 

respectively  equal  but  the  included  angles  unequal, 

(e)  chords  of  equal  circles  subtending  unequal  arcs, 

(/)   chords  of  equal  circles  unequally  distant  from  the  centers. 

EXERCISES.    SET  CI.    INEQUALITY  OF  SECTS 

1405.  At  B,  C,  and  D  are  points  taken  in  succession  on  a  semi- 
circumference  and  arc  AC  is  greater  than  arc  BD.    Prove  that 
chord  A  B  is  greater  than  chord  CD. 

1406.  Two  chords  drawn  from  a  point  in  the  circumference  are 
unequal  if  they  make  unequal  angles  with  the  radius  drawn  from 
that  point.    Which  of  the  chords  is  the  greater? 

1407.  Two  chords  drawn  through  an  interior  point  are  unequal 
if  they  make  unequal  angles  with  the  radius  drawn  through  that 
point.    Which  is  the  greater  one? 


344  PLANE  GEOMETRY 

1408.  If  two  unequal  chords  be  produced  to  meet,  the  secants 
thus  formed  are  unequal. 

1409.  In  the  equilateral   triangle  ABC,  side  BC  is  produced  to 
D,  and  DA  is  drawn.    Prove  that  BD>AD. 

1410.  If  the  bisector  of  the  right  angle  A  of  a  right  isosceles 
triangle  BAC  cuts  BC  in  D  and  is  produced  to  K  so  that  DK=AD, 
then  KC<AK;  also  than  BC. 

VH.  ANGLES  UNEQUAL. 
Show  that  they  are 

(a)  one  exterior  and  one  non-adjacent  interior  angle  of  a 

triangle, 
(6)  opposite  the  unequal  sides  of  a  triangle, 

(c)  the  angles  opposite  the  third  sides  of  two  triangles  which 

have  two  sides  respectively  equal  but  the  third  sides 
unequal, 

(d)  measured  by  unequal  arcs. 

EXERCISES.    SET  CII.    INEQUALITY  OF  ANGLES 

1411.  If  A  is  the  vertex  of  an  isosceles  triangle  ABC  and  the  leg 
AC  is  produced  to  point  D  and  DB  drawn,  prove  that  <£ABD> 


1412.  If  the  side  BC  of  the  square  ABCD  is  produced  to  P  and 
P  is  joined  with  A,  prove  that  <£APB<  $BAP. 

1413.  The  angle  formed  by  two  tangents  is  equal  to  twice  the 
angle  between  the  chord  of  contact  and  the  radius  drawn  to  a 
point  of  contact. 

1414.  If  the  diagonals  of  quadrilateral  ABCD  bisect  each  other 
at  P,  and  side  BC  is  longer  than  side  AB,  prove  that 


1415.  In  the  quadrilateral   MNRS,  MS  is  the  longest  and 
NR  the  shortest  side.     Prove  that  <£MNR>  $MSR',   also  that 


1416.  If  in  parallelogram  ABCD,  side  BOAB,  and  the  diago- 
nals intersect  in  P,  prove  that  <£BPC>  %BPA. 

TRIANGLES  SIMILAR. 
Show  that  they  have 

(a)  a  center  of  similitude, 

(6)  two  angles  respectively  equal, 


SUMMARIES  AND  APPLICATIONS  345 

(c)  two  pairs  of  sides  proportional  and  the  included  angles 

equal, 
or,  that 

(d)  their  sides  bear  a  constant  ratio, 

(e)  they  are  triangles  formed  by  homologous  diagonals  of 

similar  polygons, 

(/)   they  are  triangles  formed  by  the  altitude  upon  the  hypo- 
tenuse of  a  right  triangle. 

EXERCISES.    SET  GUI.    SIMILARITY  OF  TRIANGLES 

1417.  If  through  the  point  of  contact  of  two  tangent  circles 
three  secants  are  drawn,  cutting  the  circumferences  in  A,  B,  C, 
and  A i,  BI,  Ci,  respectively,  then  triangles  ABC  and  AiBiCi  are 
similar. 

1418.  If  from  the  point  P  outside  a  circle  two  secants  are  drawn 
to  meet  the  circumference  in  B  and  C,  and  D  and  E,  respectively, 
the  triangles  PBE  and  PCD  are  similar. 

1419.  If  the  bisector  AD  of  <£A  in  the  inscribed  triangle  ABC 
is  produced  to  meet  the  circumference  in  E,  then  triangles  ADD 
and  A  EC  are  similar. 

1420.  If  two  chords,  AB  and  CD,  intersect  in  E,  the  triangles 
A  EC  and  BED  are  similar. 

1421.  If  the  altitudes  AD  and  BE  of  triangle  ABC  intersect 
at  F,  triangles  AFE  and  BFD  are  similar. 

1422.  A  D  and  BE  are  two  altitudes  of  triangle  ABC.   Prove  that 
triangles  CDE  and  CBA  are  similar. 

IX.  SECTS  PROPORTIONAL. 
Show  that  they  are 

(a)  homologous  parts  of  similar  polygons, 
(6)  sects  of  sides  of  a  triangle  cut  off  by  line  parallel  to  third 
side, 

(c)  sects  of  transversals  cut  off  by  parallels, 

(d)  two  sides  of  a  triangle  and  the  sects  of  the  third  side  made 

by  the  bisector  of  the  opposite  (1)  interior  and  (2) 
exterior  angle, 

(e)  the  altitude  upon  the  hypotenuse  and  the  sects  of  the 

hypotenuse  made  by  it,  or  a  leg,  hypotenuse,  and  the 
projection  of  the  leg  upon  the  hypotenuse. 


346  PLANE  GEOMETRY 

EXERCISES.    SET  CIV.    PROPORTIONALITY  OF  SECTS 

1423.  Triangles  are  similar  if  an  angle  of  one  is  equal  to  an  angle 
of  another  and  the  altitudes  drawn  from  the  vertices  of  the  other 
angles  are  proportional. 

1424.  If  two  circles  are  tangent  externally,  and  through  the 
point  of  contact  a  secant  is  drawn,  the  chords  formed  are  propor- 
tional to  the  radii. 

1425.  If  C  is  the  mid-point  of  the  arc  AB,  and  a  chord  CD  meets 

ftp     C1  A 
the  chord  AB  in  E,  then  ^-r  =  ~=  • 

1426.  The  diagonals  of  any  trapezoid  divide  each  other  in  the 
same  ratio. 

X.  PRODUCTS  OF  SECTS  EQUAL. 

Show  that  the 

(a)  factors  are  sides  of  similar  polygons, 

(6)  sects  are  sects  of  chords  intersecting  inside  a  circle, 

(c)   factors  of  one  product  are  the  tangent  and  the  factors  of 

the  other  are  the  entire  secant  from  the  same  point  and  its 

external  sect. 

EXERCISES.    SET  CV.    EQUALITY  OF  PRODUCTS  OF  SECTS 

1427.  If  from  any  point  E  in  the  chord  A B  the  perpendicular 

EC  be  drawn  upon  the 
diameter  AD,  then  ~AC  X 
AD=ABXAE. 

1428.  If  in  the  triangle 
ABC  the  altitudes  AD  and 
BE  meet  in  Fjbhen  BD  X 
DC=DFXAD. 

1429.  In  the  same  diagram,  ~&D  XAC=BFXAD. 

1430.  If  a  chord  be  bisected  by  another,  either  sect  of  the  first 
is  a  mean  proportional  between  the  sects  of  the  other. 

The  preceding  summary  is  incomplete.  The  idea  contained  in 
it  may  be  developed  by  asking  the  pupil  to  make  similar  lists  of 
methods  of  establishing  such  geometric  relations  as  the  following: 
inequality  of  arcs,  conditions  under  which  a  quadrilateral  is  a 
parallelogram,  sums  of  sects  unequal,  differences  of  products 
unequal,  lines  concurrent,  points  collinear,  points  concyclic. 


SUMMARIES  AND  APPLICATIONS  347 

EXERCISES.    SET  CVI.     MISCELLANEOUS 

1431.  Draw,  through  a  given  point,  a  secant  from  which  two 
equal  circles  shall  cut  off  equal  chords. 

1432.  In  a  right  isosceles  triangle  the  hypotenuse  of  which  is 
10  in.,  find  the  length  of  the  projection  of  either  arm  upon  the 
hypotenuse. 

1433.  Find  the  projection  of  one  side  of  an  equilateral  triangle 
upon  another  if  each  side  is  6  in. 

1434.  If  two  sides  of  a  triangle  are  10  and  12,  and  their  included 
angle  is  120°,  what  is  the  value  of  the  third  side? 

1435.  If  two  sides  of  a  triangle  are  12  and  16,  and  their  included 
angle  is  45°,  find  the  third  side. 

1436.  Assuming  the  diameter  of  the  earth  to  be  8000  mi.,  how 
far  can  you  see  from  the  top  of  a  mountain  a  mile  high? 

1437.  Write  the  formula  involving  the  median  to  6,  to  c. 

1438.  If  the  sides  of  a  triangle  ABC  are  5,  7,  and  8,  find  the 
lengths  of  the  three  medians. 

1439.  If  the  sides  of  a  triangle  are  12,  16,  and  20,  find  the 
median  to  side  20.    How  does  it  compare  in  length  with  the  side 
to  which  it  is  drawn?    Why? 

1440.  In  triangle  ABC,  a  =  16,  6  =  22,  and  mc=17.    Find  c. 

1441.  In  a  right  triangle,  right-angled  at  C,  rac  =  8-|;  what  is  c? 
Find  one  pair  of  values  for  a  and  b  that  will  satisfy  the  conditions 
of  the  problem. 

1442.  If  the  sides  of  a  triangle  are  7,  8,  and  10,  is  the  angle 
opposite  10  obtuse,  right,  or  acute?    Why? 

1443.  Draw  the  projections  of  the  shortest  side  of  a  triangle 
upon  each  of  the  other  sides  (a)  in  an  acute  triangle  (6)  in  a  right 
triangle,  (c)  in  an  obtuse  triangle.    Draw  the  projections  of  the 
longest  side  in  each  case. 

1444.  Two  sides  of  a  triangle  are  8  and  12  in.  and  their  included 
angle  is  60°.    Find  the  projection  of  the  shorter  upon  the  longer. 

1445.  In  Ex.  1444  find  the  projection  of  the  shorter  side  upon  the 
longer  if  the  included  angle  is  30°;  45°. 

1446.  Write  the  formula  for  the  projection  of  a  upon  b. 

1447.  In  triangle  ABC,  a  =  15,  6  =  20,  c  =  25;  find  the  projection 
of  b  upon  c.    Is  angle  A  acute,  right,  or  obtuse? 


348  PLANE  GEOMETRY 

1448.  If  the  altitude  upon  the  hypotenuse  of  a  right  triangle 
divides  the  hypotenuse  in  extreme  and  mean  ratio,  the  smaller 
arm  is  equal  to  the  non-adjacent  sect  of  the  hypotenuse. 

1449.  If  two  circles  are  tangent  externally,  their  common  ex- 
ternal tangent  is  a  mean  proportional  between  their  diameters. 

1450.  The  sides  of  a  triangle  are  13,  17,  19.    Find  the  lengths 
of  the  sects  into  which  the  angle  bisectors  divide  the  opposite  sides. 

1451.  The  angles  of  a  triangle  are  30°,  60°,  90°.    Find  the  lengths 
of  the  sects  into  which  the  angle  bisectors  divide  the  opposite  sides, 
if  the  hypotenuse  is  10. 

1452.  Find  the  sum  of  (a)  the  acute  angles  of  a  starred  pentagon, 
(6)  all  the  interior  angles. 

1453.  The  difference  of  the  squares  of  two  sides  of  a  triangle  is 
equal  to  the  difference  of  the  squares  of  the  sects  made  by  the 
altitude  upon  the  third  side. 

1454.  The  perpendicular  erected  at  the  mid-point  of  the  base  of 
an  isosceles  triangle  passes  through  the  vertex  and  bisects  the 
angle  at  the  vertex. 

Construct  a  right  triangle,  having  given: 

1455.  The  hypotenuse  and  one  side. 

1456.  One  side  and  the  altitude  upon  the  hypotenuse. 

1457.  The  median  and  the  altitude  upon  the  hypotenuse. 

1458.  The  hypotenuse  and  the  altitude  upon  the  hypotenuse. 

1459.  The  radius  of  the  inscribed  circle  and  one  side. 

1460.  The  radius  of  the  inscribed  circle  and  an  acute  angle. 

1461.  The  hypotenuse  and  the  difference  between  the  arms. 

1462.  The  hypotenuse  and  the  sum  of  the  arms. 

1463.  The  sides  of  a  triangle  are  6,  7,  and  8  ft.    Find  the  areas 
of  the  two  parts  into  which  the  triangle  is  divided  by  the  bisector 
of  the  angle  included  by  6  and  7. 

1464.  The  square  constructed  upon  the  sum  of  two  sects  is  equal 
to  the  sum  of  the  squares  constructed  upon  these  two  sects,  in- 
creased by  twice  the  rectangle  of  the  sects. 

1465.  The  square  constructed  upon  the  difference  of  two  sects 
is  equal  to  the  sum  of  the  squares  constructed  upon  these  sects, 
diminished  by  twice  their  rectangle.- 


SUMMARIES  AND  APPLICATIONS 


349 


0 


1466.  The  difference  between  the  squares  constructed  upon  two 
sects  is  equal  to  the  rectangle  of  their  sum  and  their  difference. 

1467.  A    straight    rod    moves    so 
that  its  ends  constantly  touch  two 
fixed  rods  perpendicular  to  each  other. 
Find  the  locus  of  its  mid-point. 

1468.  If  a  quadrilateral  has  each 
side  tangent  to  a  circle,  the  sum  of  one 
pair  of  opposite  sides  equals  the  sum 
of  the  other  pair. 

1469.  Analysis  of  the  regular  inscribed  hexagon — prove  that: 
(a)  Three  of  the  diagonals  are  diameters. 

(6)  The  perimeter  contains  three  pairs  of  parallel  sides. 

(c)  Any  diagonal  which  is  a  diameter  divides  the  hexagon  into 
two  isosceles  trapezoids. 

(d)  Radii  drawn  to  the  alternate  vertices  divide  the  hexagon 
into  three  congruent  rhombuses. 

(e)  The    diagonals    joining    the    alternate    vertices    form    an 
equilateral  triangle  of  which  the  area  equals  one-half  that  of  the 
hexagon. 

(/)  The  diagonals  joining  the  corresponding  extremities  of  a 
pair  of  parallel  sides  of  the  hexagon  form  with  these  sides  a 
rectangle. 

1470.  Inscribe  in  a  given  circle  a  regular  polygon  similar  to 
a  given  regular  polygon. 

1471.  Construct  the  following  angles:  60°,  30°,  72°,  18°,  24°,  42°. 

1472.  Construct  A  ABC  given  <£B,  b,  and  mb. 

1473.  Construct  a  triangle,  having  given  the  base,  the  altitude, 
and  the  angle  at  the  vertex. 

1474.  (a)  Construct  a  square  that  shall  be  to  a  given  triangle  as 
5  is  to  4. 

(6)  Construct  a  square  that  shall  be  to  a  given  triangle  as  m  is 
to  n,  when  m  and  n  are  two  given  sects. 

1475.  The  diagonals  of  a  regular  pentagon  divide  each  other 
into  extreme  and  mean  ratio. 

1476.  If  the  sect  I  is  divided  internally  in  extreme  and  mean 
ratio;  and  if  s  is  its  greater  sect;  what  is  the  value  of  s  in  terms  of  /? 


350  PLANE  GEOMETRY 

1477.  A  sect  10  in.  long  is  divided  internally  *m  extreme  and 
mean  ratio.    Find  the  lengths  of  its  sects. 

1478.  A  sect  8  in.  long  is  divided  externally  in  extreme  and  mean 
ratio.    Find  the  length  of  its  longer  sect. 

1479.  Experience  has  shown  that  a  book,  photograph,  or  other 
rectangular  object  is  most  pleasing  to  the  eye  when  its  length  and 
width  are  obtained  by  dividing  the  semiperimeter  into  extreme  and 
mean  ratio.    Find  to  the  nearest  integer  the  width  of  such  a  book 
where  its  length  is  8  in. 

1480.  If  the  bisector  of  an  inscribed  angle  be  produced  until  it 
meets  the  circumference,  and  through  this  point  of  intersection  a 
chord  be  drawn  parallel  to  one  side  of  the  angle,  it  is  equal  to  the 
other  side. 

1481.  If  from  the  extremities  of  a  diameter  perpendiculars  be 
drawn  upon  any  chord  (produced  if  necessary),  the  feet  of  the 
perpendicular  are  equidistant  from  the  center. 

1482.  Find  the  locus  of  points,  the  distances  of  which  from  two 

intersecting  lines  L  and  L\  are  as  —  . 

IV 

The  locus  consists  of  two  straight  lines.    Draw  parallels  to  L  and 
Li,  such  that  their  distances  from  L  and  LI  respectively  shall  be 

as—  ;  these  parallels  will  intersect  in  points  belonging  to  the 


required  locus.    Special  case  :    <£  LLi  =  60°,  m  —  2,  n  =  1  . 

1483.  Between  the  sides  of  a  given  angle  a  series  of  parallels 
are  drawn;  find  the  locus  of  points  which  divide  these  parallels  in 

the  ratio  —  .    Special  case  :  m  =  4,  n  =  1  . 

1484.  Construct  a  triangle  equal  to  the  sum  of  two  given 
triangles. 

1485.  Construct  a  triangle  equal  to  the  difference  of  two  given 
parallelograms. 

1486.  Construct  a  regular  pentagon,  given  one  of  the  diagonals. 

1487.  Prove  that  the  following  solution  of  the  problem  to  divide 
a  given  circle  into  any  number  of  equal  parts  (say  3),  by  drawing 
concentric  circles,  is  correct. 


SUMMARIES  AND  APPLICATIONS  351 

Trisect  the  radius  at  M  and  N.    Draw  semicircle  on  A  0  as  diam- 
eter.    Erect  perpendiculars  MB  and 
NC  meeting  semicircle  at  B  and   C. 
With  centers  at  0  and  radii  OB  and 
OC,  draw  circles. 

1488.  Transform  a  rectangle  into  a 
parallelogram,  having  given  a  diagonal. 

1489.  Transform    a    given    triangle 
into  a  right  triangle  containing  a  given 
acute  angle. 

1490.  The  sides  of  a  triangle  are  8,  15,  and  17.    Find  the  radius 
of  the  inscribed  circle. 

1491.  The  sum  of  the  squares  on  the  sects  of  two  perpendicular 
chords  is  equal  to  the  square  of  the  diameter  of  the  circle. 

If  A  B,  CD  are  the  chords,  draw  the  diameter  BE,  and  draw  AC, 
ED,  BD.    Prove  that  AC=ED. 

1492.  Calculate  the  lengths  of  the  common  external  and  internal 
tangents  to  two  circles  whose  radii  are  16  and  12  units  respectively, 
and  whose  line  of  centers  is  40. 

1493.  Describe  a  circle  whose  circumference  is  equal  to  the 
difference  of  two  circumferences  of  given  radii. 

1494.  Construct  a  circle  equal  to  three-fifths  of  a  given  circle. 

1495.  Construct  a  circle  equal  to  three  times  a  given  circle. 

1496.  Construct  a  semicircle  equal  to  a  given  circle. 

1497.  Construct  a  circle  equal  to  the  area  bounded  by  two  con- 
centric circumferences. 

Required  to  circumscribe  about  a  given  circle  the  following 
regular  polygons: 

1498.  Triangle.         1499.  Quadrilateral.   1500.  Hexagon. 
1501.  Octagon.          1502.  Pentagon.          1503.  Decagon. 

1504.  Construct  a  square  equal  to  a  given  (a)  parallelogram, 
(6)  triangle,  (c)  polygon. 

1505.  Construct  a  triangle  similar  to  a  given  triangle  and  equal 
to  another  given  triangle. 

1506.  Construct  a  polygon  similar  to  a  given  polygon  and  having 
a  given  ratio  to  it. 


352 


PLANE  GEOMETRY 


1507.  Construct  x= 


ab 


1508.  Construct  x= \/a2+62  -ab. 

1509.  Construct  the  roots  of  xz+ax-\-b=0. 

1510.  Givenjbhe  sects  a,  6,  c,  construct  sect  x  if: 

(a)  z=\/3a6.  (6)  x=Va2 -bz.  (c)  z= 

1511.  Transform  a  given  triangle  into  an  equilateral  triangle. 

1512.  Transform  a  given   parallelogram  into   an  equilateral 
triangle. 

1513.  Construct  an  equilateral  triangle  equal  to  one-half  a  given 
square. 

1514.  Transform  a  triangle  into  a  right  isosceles  triangle. 

1515.  Divide  a  given  sect  into  two  sects  such  that  one  is  to  the 
given  sect  as  \/2  is  to  A/5? 

1516.  In  a  circle  a  line  EF  is  drawn  per- 
pendicular to  a  diameter  AB,  and  meeting 
it  in  G.  Through  A  any  chord  AD  is  drawn, 
meeting  EF  in  C.  Prove  that  the  product 
AD  XAC  is  constant,  whatever  the  direc- 
tion of  AD. 

Draw  BD  and  compare  the  A  ACG, 
ADB.  Is  the  theorem  also  true  when  G 
lies  outside  the  circle? 

1517.  If  two  sects  OA,  OB,  drawn_through  a  point  0  are  divided 
in  C,  D,  respectively,  eo  that  OA  XOC 

=OBXOD,  a  circle  can  be  described 
through  the  points  A,  B,  C,  D. 

Show  that  A  DAO,  CBO  are  sim- 
ilar, and  the  $?DAC  and  CBD  equal.  0 
Therefore,  if  a  segment  is  described 
upon    CD  capable  of  containing  the 
-&DAO,  the  arc  of  this  segment  will  pass  through  B. 

1518.  Divide  a  quadrilateral  into  four  equal  parts  by  lines  drawn 
from  a  point  in  one  of  its  sides. 

1519.  Draw  a  common  secant  to  two  given  circles  exterior  to 
each  other,  such  that  the  intercepted  chords  shall  have  the  given 
lengths  a  and  6. 


SUMMARIES  AND  APPLICATIONS 


353 


D 


H        C 

as  to  fulfill  the 
AH  and  AK  are 


1620.  Divide  quadrilateral  ABCD 
into  three  equal  parts  by  straight 
lines  passing  through  A. 

(a)  Transform  ABCD  into  AADE. 

Divide  AADE  into  the  three  equal 
parts  ADF,  AFG,  and  AGE. 

As  the  last  two  parts  do  not  lie 
entirely  in  the  given  quadrilateral 

drawGtf  ||CA. 

Then  AFG  =  AFCH,  and  AF  and 
AH  are  the  required  lines. 

Or   (6)  Trisect  DB.    Draw  AF, 
AE,  CF,  and  CE. 

Then  the  broken  lines  AFC  and 
AEC  divide  the  figure   into   three 
equal    parts.    To    transform    these    parts  so 
conditions,  draw  FH  and  EK  parallel  to  AC. 
the  required  lines. 

1521.  If  the  diagonals  of  a  trapezoid  are  equal,  the  trapezoid  is 
isosceles. 

1622.  If  the  altitude  BD  of  AABC  is  intersected  by  another 

altitude  in  G,  and  EH  and  #F 
are  perpendicular  bisectors  of  AC 
and  CB~,  prove  BG  =  2HE. 

1623.  The  line  joining  the  point 
of  intersection  of  the  altitudes  of 
a  triangle  and  the  point  of  inter- 
section of  the  three  perpendicular 
bisectors,  cuts  off  one-third  of  the  B 
corresponding  median. 

1524.  The  points  of  intersection  of 
the  altitudes,  the  medians,  and  the 
perpendicular  bisectors  of  a  triangle 
lie  in  a  straight  line. 

1525.  If,  through  the  points  of  intersection  of  two  circumfer- 
ences, parallels  be  drawn  terminated  by  the  circumferences,  they 
are  equal. 

23 


F 


O 


354 


PLANE  GEOMETRY 


1526.  If  from  any  point  in  the  circumference  of  a  circle  chords 
be  drawn  to  the  vertices  of  an  inscribed  equilateral  triangle,  the 
longest  chord  equals  the  sum  of  the  smaller  chords. 

1527.  Triangles  are  similar  if  two  sides  and  the  radius  of  the 
circumscribed  circle  of  one  are  proportional  to  the  homologous 
parts  of  another. 

1528.  Construct  a  square  that  shall  be  to  a  given  triangle  as  5 
is  to  4. 

1529.  Construct  a  square  that  shall  be  to  a  given  triangle  as 
m  is  to  n,  when  m  and  n  are  two  given  sects. 

1530.  If  in  a  circle  a  regular  decagon  and  a  regular  pentagon  be 
inscribed,  the  side  of  the  decagon  increased  by  the  radius  is  equal 
to  twice  the  apothem  of  the  pentagon. 

1531.  If  from  a  point  0,  OA,  OB,  OC,  and  OD  are  drawn  so  that 
the  <£  AOB  is  equal  to  the  <£BOC,  and  the  <&BOD  equal  to  a  right 
angle,  any  line  intersecting  OA,  OB,  OC,  and  OD  is  divided 
harmonically. 

1532.  Inscribe  in  a  given  circle  a  regular  polygon  of  n  sides, 
n  being  any  whole  number. 

The  following  construction  is  found  in  most  cases  to  be  suffi- 
ciently exact  for  practical  purposes: 

Divide  the  diameter  AB  into  n 
equal  parts   (in  the  figure  n  =  7). 
Draw  the  radius  CD  A.AB,  produce 
CB  to  E,  and  CD  to  F,  making  BE 
and  DF  each  equal  to  one  of  the 
parts  of  the  diameter ;   draw  EF, 
cutting  the  circle  for  the  first  time 
in  G.    Then  the  line  GH  joining  G 
and  the  third  point  of  division  of  AB,  counting  from  B  will  be 
very  nearly  equal  to  one  side  of  the  inscribed  polygon  of  n  sides. 
For  n  =  3  and  n  =  4,  this  construction  is  impossible;  for  n  =  5  it 
is  useless,  on  account  of  its  inaccuracy;  but  for  n>5  it  gives  a 
very  close  approximation  to  the  exact  value  of  the  side  required. 
Inscribe  a  hexagon,  a  heptagon,  an  octagon,  a  nonagon,  and 
a  decagon. 


E 


SUMMARIES  AND  APPLICATIONS 


355 


1533.  Draw  a  line  meeting  the  sides  CA,  CB,  of  AABC  in  D, 
E,  respectively,  so  that: 

(a)  DE  ||  AB,  BE=ED.    N.B.  Bisect 


(6)  DE  ||  AB,  DE^AD+BE. 
(c)  DE=CD^BE.     (See  figure.) 
Analysis:    Suppose  the  problem  solved,  and 
draw  BD.    The  A  BDE  and  DCE  are  isosceles, 


whence  %.DBE  =  ^BDE=^DEC  =  ^DCE,  which  is  known.    This  deter- 
mines the  point  D,  and  E  is  easily  found,  since  DE^DC. 

Examine  this  problem  for  the  special  cases 
when  <£AC#  =  90°,  and  when  <£ACE  =  12Q°. 

1534.  Draw  through  a  given  point  P  in  the 
arc  subtended  by  a  chord  AB  a  chord  which 
shall  be  bisected  by  AB. 

r(j      On  radius  OP  take  CD  equal  to  CP.    Draw 
DE\\BA. 

1535.  Through  a  given  point  P  inside  a 
given  circle  draw  a  chord  AB  so  that  the  ratio 
AP_m 
BP 


n 

Draw  OPC  so^that  157;=  ~-    Draw   CA   equal   to 
proportional  to  n,  m,  and  the  radius  of  the  circle. 

1536.  Draw    through    one    of    the 
points  of  intersection  of  two  circles  a 
secant  so  that  the  two  chords  that  are 
formed  shall  be  in  the    given    ratio 
m  to  n. 

1537.  Draw  a  common  tangent  to 
two  given  circles : 

(a)  Non-intersecting  circles 
(1)  unequal,  (2)  equal. 

(b)  Intersecting 

(1)  unequal,  (2)  equal. 

(c)  Tangent  (internally  and  externally) 

(1)  unequal,  (2)  equal. 


the   fourth 


356 


PLANE  GEOMETRY 


1638.  Find  the  shortest  path  from  P  to  Pi  which  shall  touch 
a  given  line,  if  P  and  Pi  are  two  given  points  on  the  same  side 
of  the  line. 

1539.  Given  points  P,  Pi,  Pz',  through  P  draw  a  line  which  shall 
be  equidistant  from  Pi  and  P%. 

1540.  Given  P,  Pi,  Pz',  through  P  draw  a  line  so  that  the  dis- 
tances from  P  to  the  feet  of  the  perpendiculars  dropped  from  PI, 
P2  to  the  line,  shall  be  equal. 

1541.  Find  the  direction  in  which  a  billiard  ball  must  be  shot 

from  a  given  point  on  the  table, 
so  as  to  strike  another  ball  at  a 
given  point  after  first  striking 
one  side  of  the  table.  (The  angle 
at  which  the  ball  is  reflected  from 
a  side  is  equal  to  the  angle  at 
which  it  meets  the  side,  that  is, 


D    E       JB 


Suggestion:  Construct  BE  _L  that  side  of  the  table  which  the  ball  is  to 
strike,  and  make  ED=BE. 

1542.  The  same  as  the  preceding  problem,   except  that  the 
cue  ball  is  to  strike  two  sides  of  the 

table  before  striking  the  other  ball. 
(Fig.  1.) 

Suggestion:   BiEi  =#iDi,  Dtf  =OT,. 

1543.  Solve  Ex.  1542  if  the  cue  ball 
is  to  strike  three  sides  before  striking 
the  other  ball,  also  if  it  is  to  strike  all 
four  sides.    (Fig.  2.) 

1544.  A  billiard  ball 
is  placed  at  a  point  P 

on  a  billiard  table.  In  what  direction  must  it  be 
shot  to  return  to  the  same  point  after  hitting  all 
four  sides? 

Suggestion:  (a)  Show  that  the  opposite  sides  of  the  quadrilateral  along 
which  the  ball  travels  are  parallel.  (&)  If  the  ball  is  started  parallel  to  a  diag- 
onal of  the  table,  show  that  it  will  return  to  the  starting  point. 


FIG.  2. 


SUMMARIES  AND  APPLICATIONS  357 

1545.  Show  that  in  the  preceding  problem  the  length  of  the  path 
traveled  by  the  ball  is  equal  to  the  sum  of 

the  diagonals  of  the  table. 

1546.  Show  that  in  the  diagram  BC  =  i$ 
and  OR  =  iio  and  QC  =  z6where  OC±AD,~OD 


=  ±OC,  and  BD  =DC. 

d!547.  Describe  a  circle  the  ratio  of  whose  area  to  that  of  a 
given  circle  shall  be  equal  to  the  given  ratio  m  to  n. 

d!548.  In  an  inscribed  quadrilateral  the  product  of  the  diagonals 
is  equal  to  the  sum  of  the  products  of  the  opposite  sides.  (Ptolemy.) 

d!549.  Find  a  point  such  that  the  perpendiculars  from  it  to 
the  sides  of  a  given  triangle  shall  be  in  the  ratio  p  to  q  to  r. 

d!550.  Find  a  point  within  a  triangle  such  that  the  sects  joining 
the  point  with  the  vertices  shall  form  three  triangles,  having  the 
ratio  3  to  4  to  5. 

d!551.  Given  a  circle  and  its  center;  find  the  side  of  an  inscribed 
square  by  means  of  the  compasses  alone.  ("  Napoleon's  Problem.") 

d!552.  Bisect  a  trapezoid  by  a  line  parallel  to  the  bases. 

d!553.  The  feet  of  the  perpendiculars  dropped  upon  the  sides 
of  a  triangle  from  any  point  in  the  circumference  of  the  circum- 
scribed circle  are  collinear.  ("  Simpson's  Line.") 

d!554.  The  points  A,B,C,  D,  are  collinear.  Find  the  locus  of  a 
point  P  from  which  the  sects  AB  and  CD  subtend  the  same  angle. 

d!555.  Transform  a  given  triangle  into  one  containing  two  given 
angles. 

d!556.  Transform  a  given  triangle  into  an  isosceles  triangle, 
having  a  given  vertex  angle. 

Hint:  Construct  a  A  similar  to  a  given  isosceles  A  and  equal  to  a  given 
A,  or  transform  a  A  into  one  containing  two  given  2$.s. 

d!557.  Construct  an  equilateral  triangle  that  shall  be  to  a  given 
rectangle  as  4  is  to  5. 

d!558.  In  a  given  triangle,  ABC,  inscribe  a  parallelogram  similar 
to  a  given  parallelogram,  so  that  one  side  lies  in  AB,  and  the  other 
two  vertices  lie  in  BC  and  AC  respectively. 

d!559,  Divide  a  pentagon  into  four  equal  parts  by  lines  drawn 
through  one  of  its  vertices. 


358 


PLANE  GEOMETRY 


dl560.  Right  triangles  are  similar  if  the  hypotenuse  and  an 
'arm  of  one  triangle  are  proportional  to  the  hypotenuse  and  an  arm 

of  another. 

d!561.  If  from  a  point  A  two  equal 
tangents,  AB  and  AC,  are  drawn  to 
two  circles,  0  and  Oi,  and  AD  is  per- 

2         2 

pendicular  to  00\,  then  OD  —0\D  = 
0g2  _ojf. 

d!662.  Conversely,  if.  in  the  same 

2  2  2  — .  2 

diagram,  D  is  taken  so  that  OD  -OJ)  =  OB  -  OiC  ,  then  the 
tangents  drawn  from  any  point  in  the  perpendicular,  AD,  to  the 
circles  are  equal. 

d!663.    Divide   a 
trapezoid  into    two  *** 
similar  trapezoids  by 
a  line  parallel  to  the 
bases. 

d!664.  Through  P 
secants  are  drawn  to  a  circle  0;  find  the  locus  of  points  which 

divide  the  entire  secants  in  the  ratio  — 

n 

F  d!565.  Through  a  fixed  point 

F  a  secant  is  drawn  to  a  given 
circle,  and  through  its  intersec- 
tions A,  B  with  the  circumfer- 
ence tangents  are  drawn  inter- 
secting in  a  point  P.  If  the  se- 
cant revolves  about  F,  find  the 
locus  of  P. 

d!566.  Find  the  locus  of  the 

vertex  of  a  triangle,  having  given  the  base  and  the  ratio  of  the 

other  two  sides. 

CONSTRUCTIONS  LEADING  TO  THE  PROBLEM  OF  APOLLONIUS 

(200  B.C.) 

d!567.  Construct  a  circle  which  shall  pass  through  two  given 
points  and  be  tangent  to  a  given  line. 


SUMMARIES  AND  APPLICATIONS 


359 


d!668.  Construct  a  circle  which  shall  pass  through  two  given 
points  and  be  tangent  to  a  given  circle. 

d!569.  Construct  a  circle  which  shall  pass  through  a  given  point 
and  be  tangent  to  two  given  lines. 

dl670.  Construct  a  circle  which  shall  pass  through  a  given  point 
and  be  tangent  to  a  given  line  and  to  a  given  circle. 

d!571.  Construct  a  circle  which  shall  pass  through  a  given  point 
and  be  tangent  to  two  given  circles. 

d!572.  Construct  a  circle  which  shall  be  tangent  to  three  given 
lines. 

d!673.  Construct  a  circle  which  shall  be  tangent  to  two  given 
lines  and  to  a  given  circle. 

d!574.  Construct  a  circle  which  shall  be  tangent  to  a  given  line 
and  to  two  given  circles. 

d!575.  Construct  a  circle  which  shall  be  tangent  to  three  given 
circles.  ("  Problem  of  Apollonius.") 

THE  TRIANGLE  AND  NINE  OF  ITS  CIRCLES 

The  three  escribed  circles  are  represented  by  arcs;  the  other 
six  by  centers.  Prove  the  construction  of  the  nine  circles.  Find 
three  others. 

Circles.  Centers.  Radii. 

Circumscribed,                      1  IA 

Inscribed,                              2  Perp.Jrom  2 

Nine  points,                          3  34 

Pedal  (3),  4,  5,  6  40~,  50~,  60 
C  is  the  centroid;  and  0,  the 


d!576.  The  circumscribed 
circle  bisects  the  straight  lines 
joining  the  center  of  the  in- 
scribed circle  with  the  centers 
of  the  escribed  circles. 

d!577.  Each  vertex  of  the  tri- 
angle is  collinear  with  centers 
of  two  of  the  escribed  circles. 

d!578.  The  center  of  the  inscribed  circle  is  collinear  with  the 
center  of  any  escribed  circle  and  the  opposite  vertex. 


360 


PLANE  GEOMETRY 


d!579.  Each  center  of  the  inscribed  or  the  escribed  circles  is  the 
orthocenter  of  the  triangle  having  the  other  three  centers  as  its 
vertices. 

d!580.  The  four  circles,  each  of  which  passes  through  three  of 
the  centers  of  the  escribed  and  inscribed  circles,  are  equal. 

d!581.  The  three  circles,  the  circumference  of  each  of  which 
passes  through  the  extremities  of  any  side  of  a  triangle  and  the 
orthocenter,  equal  one  another. 

Nineteen  circles  in  all  have  been  mentioned. 

THE  NINE-POINTS  CIRCLE 

The  orthocenter  is  the  point  at  which  the  altitudes  of  a  triangle 
meet. 

The  centroid  of  any  triangle  is  the  point  at  which  the  medians  of 
the  triangle  meet. 

1582.  The  mid-points  of  the  sides  of  a  triangle  are  concyclic  with 
the  feet  of  the  perpendicular  from  the  opposite  vertices,  and  with 
the  mid-points  of  the  sects  joining  the  orthocenter  with  the  vertices. 
(Nine-points  circle.) 

1583.  The  center  of  the  nine-points  circle  is  the  mid-point  of 
the  sect  joining  the  orthocenter  and  the  center  of  the  circumscribed 
circle. 

1584.  The  diameter  of  the  nine-points  circle  is  equal  to  the  radius 
of  the  circumscribed  circle. 

1585.  The  orthocenter  and  the  centroid  are  collinear  with  the 
centers  of  the  nine-points  and  the  circumscribed  circles. 

1586.  The  nine-points  circle  is  tangent  to  the  inscribed  and 

escribed  circles  of  a  triangle. 

Give  proofs  different  from  those  sug- 
gested in  the  text  for  the  following 
theorems: 

1587.  Theorem  216.  Suggestion:  (Fig.  1). 
Suggestion:  (Figs.  2  and  3). 


Fia.  1 

1588.  Theorem  2ld. 


FIG.  2 


Fie.  3 


SUMMARIES  AND  APPLICATIONS 


361 


1589.  Theorem  21c.    Prove  by  the  method  of  exclusion. 

1590.  Theorem  21/.    Suggestion:    (Fig.  A). 

1591.  Theorem  26a.    For  suggestions  see  Heath's  Mathematical 
Monographs,  Numbers  1  and  2. 


FIG.  B 


FIG.  A 

1592.  Theorem  266.    Suggestion:   (Fig.  B). 

1593.  Theorem  40.    Prove  by  means  of  direct  and  opposite. 

1594.  Theorem  41.    Prove  as  in  Ex.  1593. 

1595.  Theorem  53.    Suggestion:    (Fig.  1). 


FIG.  1  FIG.  2  Fio.  3 

1596.  Theorem  54.    Suggestions:    (Figs.  2  and  3). 

1597.  Theorem  55,  Cor.  1.    Suggestion:  .(Fig.  4). 

1598.  Use  the  following  analysis  to  find 
another  solution  for  Problem  21. 
Analysis:   Suppose  the  construction  completed. 

s        a 
Then  —  =  — 
a     s—a 

or      a2=s(s—  a) 
or      a2=s2— so 
or      a2 -fas =s* 

(SY_      f  s\ 
2!/  =s  ~^~\27 

which  suggests  that  (a+Tr)  is  the  hypotenuse  in  which  the  legs  are 
s  and  -n  respectively. 


FIG.  4 


CHAPTER  IX 

COLLEGE  ENTRANCE  EXAMINATIONS 

THE  UNIVERSITY  OF  CHICAGO 

EXAMINATION  FOR  ADMISSION,  JUNE,  1908 

MATHEMATICS  (2)— PLANE  GEOMETRY 
(TIME  ALLOWED — ONE  HOUR  AND  THIRTY  MINUTES.) 

[In  writing,  use  only  one  side  of  the  paper,  put  your  name  in  full  at  the 
top  of  each  sheet,  and  number  your  work  according  to  the  numbers  on  the 
printed  paper.] 

[When  required,  give  all  reasons  in  full,  and  work  out  proofs  and  problems 
in  detail.] 

State: 

(a)  At  what  school  you  studied  this  subject. 
(6)  How  many  weeks. 

(c)  How  many  recitations  per  week. 

(d)  What  textbook  you  used. 

I.  Given  two  circles  of  unequal  radii  and  lying  exterior  to  each 
other;  make  the  construction  by  which  a  straight  line  may  be 
drawn  tangent  to  both  circles  and   shall  cross  the  line  joining 
their  centers. 

II.  Prove  that  the  sum  of  the  exterior  angles  of  any  convex 
polygon,  made  by  producing  each  of  its  sides  in  consecutive  order, 
is  equal  to  four  right  angles. 

III.  In  any  triangle  PQR  perpendiculars  are  let  fall  to  the  oppo- 
site sides  from  the  vertices  P  and  Q.    Show  that  the  lines  joining 
the  feet  of  these  perpendiculars  to  the  middle  point  of  the  side 
PQ  are  equal.    [Draw  two  figures,  in  one  of  which  the  angle  at  Q 
shall  be  obtuse,  in  the  other  acute.] 

IV.  Two  fields  are  of  similar  shape,  one  having  five  times  the 
area  of  the  other,    (a)  If  they  are  both  circles  and  the  radius  of 
the  first  is  25  rods,  find   the  radius  of  the  second.     (6)  If  they 
are  both  equilateral  triangles,  and  the  side  of  the  first  is  25  rods, 
find  the  side  of  the  other. 

362 


COLLEGE  ENTRANCE  EXAMINATIONS  363 

EXAMINATION  FOR  ADMISSION,  SEPTEMBER,  1916 

MATHEMATICS  (2)— PLANE  GEOMETRY 
(TIME  ALLOWED — ONE  HOUR  AND  FIFTEEN  MINUTES.) 

fin  writing  use  only  one  side  of  the  paper,  put  your  name  in  full  at 
the  top  of  each  sheet,  and  number  your  work  according  to  the  numbers  on  the 
printed  paper.] 

[When  required,  give  all  reasons  in  full,  and  work  out  proofs  and  problems 
in  detail.] 

1.  Prove  that  a  straight  line  parallel  to  one  side  of  a  triangle 
divides  the  other  two  sides  proportionally. 

2.  Find  the  locus  of  all  points  such  that  the  two  lines  joining 
each  to  two  fixed  points  always  make  a  given  angle  with  each  other. 
What  does  the  locus  become  when  the  given  angle  is  a  right  angle? 

3.  Prove  that  in  a  triangle  with  a  given  fixed  base  the  median 
from  the  vertex  opposite  this  base  is  greater  than,  equal  to,  or  less 
than  half  the  base  according  as  the  vertical  angle  is  less  than,  equal 
to,  or  greater  than  a  right  angle. 

4.  Construct  a  circle  passing  through  a  given  point  and  tangent 
to  two  given  intersecting  straight  lines. 

5.  Two  tangents  to  a  circle  intersect  in  a  point  which  is  50  inches 
from  the  center  of  the  circle.     The  area  of  the  four-sided  figure 
formed  by  the  two  tangents  and  the  two  radii  drawn  to  the  two 
points  of  contact  is  625  square  inches.    Find  the  length  of  the  tan- 
gents and  the  radius  of  the  circle. 

6.  Show  how  to  construct  geometrically  a  square  that  shall 
contain  the  same  area  as  a  given  rectangle  whose  base  is  6  and 
whose  altitude  is  a.    Prove  the  result. 

HARVARD  UNIVERSITY 
JUNE,  1894 

(In  solving  problems  use  for  IT  the  approximate  value  3y.) 

1.  Prove  that  any  quadrilateral  the  opposite  sides  of  which  are 
equal,  is  a  parallelogram. 

A  certain  parallelogram  inscribed  in  a  circumference  has  two 


364  PLANE  GEOMETRY 

sides  20  feet  in  length  and  two  sides  15  feet  in  length;  what  are 
the  lengths  of  the  diagonals? 

2.  Prove  that  if  one  acute  angle  of  a  triangle  is  double  another, 
the  triangle  can  be  divided  into  two  isosceles  triangles  by  a  straight 
line  drawn  through  the  vertex  of  the  third  angle. 

Upon  a  given  base  is  constructed  a  triangle  one  of  the  base 
angles  of  which  is  double  the  other.  The  bisector  of  the  larger 
base  angle  meets  the  opposite  side  at  the  point  P.  Find  the  locus 
of  P. 

3.  Show  how  to  find  a  mean  proportional  between  two  given 
straight  lines,  but  do  not  prove  that  your  construction  is  correct. 

Prove  that  if  from  a  point,  0,  in  the  base,  BC,  of  a  triangle,  ABC, 
straight  lines  be  drawn  parallel  to  the  sides,  AB,  AC,  respectively, 
so  as  to  meet  AC  in  M  and  AB  in  N,  the  area  of  the  triangle  AMN 
is  a  mean  proportional  between  the  areas  of  the  triangles  BNO 
and  CMO. 

4.  Assuming  that  the  areas  of  two  parallelograms  which  have 
an  angle  and  a  side  common  and  two  other  sides  unequal,  but 
commensurable,  are  to  each  other  as  the  unequal  sides,  prove  that 
the  same  proportion  holds  good  when  these  sides  have  no  common 
measure. 

5.  Every  cross-section  of  the  train  house  of  a  railway  station 
has  the  form  of  a  pointed  arch  made  of  two  circular  arcs  the  centres 
of  which  are  on  the  ground.    The  radius  of  each  arc  is  equal  to 
the  width  of  the  building  (210  feet) ;  find  the  distance  across  the 
building  measured  over  the  roof,  and  show  that  the  area  of  the 
cross-section  is  3675  (4ir  -3\'  3)  square  feet. 

SEPTEMBER,  1894 

One  question  may  be  omitted. 

(In  solving  problems  use  for  TT  the  approximate  value  3y.) 

1.  Prove  that  any  quadrilateral  the  diagonals  of  which  bisect 
each  other  is  a  parallelogram. 

The  diagonals  of  a  parallelogram  circumscribed  about  a  circum- 
ference are  60  inches  and  80  inches  long  respectively.  How  long 
are  the  sides? 


COLLEGE  ENTRANCE  EXAMINATIONS          365 

2.  Prove  that  the  difference  of  the  angles  at  the  base  of  a  tri- 
angle is  double  the  angle  between  a  perpendicular  to  the  base 
and  the  bisector  of  the  vertical  angle. 

The  sum  of  the  base  angles  of  each  of  a  number  of  triangles  con- 
structed on  a  given  base  10  inches  long  is  150°.  What  is  the  locus 
of  the  vertices  of  these  triangles? 

3.  Show  how  to  find  a  fourth  proportional  to  three  given  lines, 
but  do  not  prove  that  your  construction  is  correct. 

One  circle  touches  another  internally  at  0,  and  a  chord  AB  of 
the  larger  circle  touches  the  smaller  one  at  C.  Prove  that  AO 
makes  with  the  common  tangent  to  the  circles  an  angle  equal  to 
ABO  and  that  CO  bisects  the  angle  AOB.  State  without  proof 
some  relation  that  exists  between  the  lines  AO,  CB,  BO,  and  AC. 

4.  Assuming  that  the  areas  of  two  rectangles  which  have  equal 
altitudes  are  to  each  other  as  their  bases  when  the  latter  are  com- 
mensurable, show  that  the  same  proportionality  exists  when  the 
bases  have  no  common  measure. 

5.  A  kite-shaped  racing  track  is  formed  by  a  circular  arc  and 
two  tangents  at  its  extremities.    The  tangents  meet  at  an  angle 
of  60°.    The  riders  are  to  go  round  the  track,  one  on  a  line  close  to 
the  inner  edge,  the  other  on  a  line  everywhere  5^  feet  outside  the 
first  line.    Show  that  the  second  rider  is  handicapped  by  about 
22  feet. 

JUNE,  1895 

One  question  may  be  omitted. 
(In  solving  problems  use  for  TT  the  approximate  value  3y.) 

1.  Prove  that  if  two  straight  lines  -are  so  cut  by  a  third  that 
corresponding  alternate-interior  angles  are  equal,  the  two  lines 
are  parallel  to  each  other. 

2.  Prove  that  an  angle  formed  by  two  chords  intersecting  within 
a  circumference  is  measured  by  one-half  the  sum  of  the  arcs  inter- 
cepted between  its  sides  and  between  the  sides  of  its  vertical  angle. 

Two  chords  which  intersect  within  a  certain  circumference 
divide  the  latter  into  parts  the  lengths  of  which,  taken  in  order, 
are  as  1,  1,  2,  and  5;  what  angles  do  the  chords  make  with  each 
other? 


PLANE  GEOMETRY 

3.  Through  the  point  of  contact  of  two  circles  which  touch  each 
other  externally,  any  straight  line  is  drawn  terminated  by  the 
circumferences;  show  that  the  tangents  at  its  extremities  are  paral- 
lel to  each  other. 

What  is  the  locus  of  the  point  of  contact  of  tangents  drawn 
from  a  fixed  point  to  the  different  members  of  a  system  of  concen- 
tric circumferences? 

4.  Prove  that,  if  from  a  point  without  a  circle  a  secant  and  a 
tangent  be  drawn,  the  tangent  is  a  mean  proportional  between 
the  whole  secant  and  the  part  without  the  circle. 

Show  (without  proving  that  your  construction  is  correct)  how 
you  would  draw  a  tangent  to  a  circumference  from  a  point 
without  it. 

5.  Prove  that  the  area  of  any  regular  polygon  of  an  even  number 
of  sides  (2n)  inscribed  in  a  circle  is  a  mean  proportional  between 
the  areas  of  the  inscribed  and  the  circumscribed  polygons  of  half 

the  number  of  sides.  If  n  be  indefinitely 
increased,  what  limit  or  limits  do  these  three 
areas  approach? 

6.  The  perimeter  of  a  certain  church  win- 
dow is  made  up  of  three  equal  semi-circum- 
ferences, the  centres  of  which  form  the  vertices 
of  an  equilateral  triangle  which  has  sides  3^ 
feet  long.  Find  the  area  of  the  window  and  the  length  of  its 
perimeter. 

SEPTEMBER,  1895 

One  question  may  be  omitted. 

(In  solving  problems  use  for  TT  the  approximate  value  3y.) 

1.  Prove  that  every  point  in  the  bisector  of  an  angle  is  equally 
distant  from  the  sides  of  the  angle.     State  the  converse  of  this 
proposition.    Is  this  converse  true? 

2.  Prove  that  an  angle  formed  by  two  secants  intersecting  with- 
out a  circumference  is  measured  by  half  the  difference  of  the  arcs 
which  the  sides  of  the  angle  intercept. 

A  certain  pair  of  secant  lines  which  intersect  without  a  circle 
divide  the  circumference  into  parts  the  lengths  of  which,  taken  in 


COLLEGE  ENTRANCE  EXAMINATIONS  367 

order,  are  to  one  another  as  1,  2,  3,  and  4.    What  angles  do  the  lines 
make  with  each  other? 

3.  Two  given  circles  touch  each  other  externally  at  the  point  Pt 
where  they  have  the  common  tangent  PC.    They  are  also  touched 
by  the  line  AB  in  the  points  A  and  B  respectively.    Show  that  the 
circle  described  on  AB  as  diameter  has  its  centre  on  PC,  and 
touches  at  P  the  straight  line  which  joins  the  centres  of  the  two 
given  circles. 

4.  Show  how  to  describe  upon  a  given  straight  line  a  segment 
which  shall  contain  a  given  angle. 

A  and  B  are  two  fixed  points  on  the  circumference  of  a  circle, 
and  PQ  is  any  diameter.  What  is  the  locus  of  the  intersection  of 
PA  and  QBt 

5.  C  is  any  point  on  the  straight  portion,  AB,  of  the  boundary 
of  a  semicircle.    CD,  drawn  at  right  angles  to  AB,  meets  the  cir- 
cumference at  D.    DO  is  drawn  to  the  centre,  0,  of  the  circle,  and 
the  perpendicular  dropped  from  C  upon  OD  meets  OD  at  E    Show 
that  DC  is  a  mean  proportional  to  AO  and  DE. 

State  the  fundamental  theorem  in  the  method  of  limits  as  used 
in  Plane  Geometry. 

6.  A  horse  is  tethered  to  a  hook  on  the  inner  side  of  a  fence 
which  bounds  a  circular  grass  plot.    His  tether  is  so  long  that  he 
can  just  reach  the  centre  of  the  plot.    The  area  of  so  much  of  the 
plot  as  he  can  graze  over  is  ^  (4?r  — 3\/3)  square  rods;  find  the 
length  of  the  tether  and  the  circumference  of  the  plot. 

JUNE,  1896 
One  question  may  be  omitted. 

(In  solving  problems  use  for  TT  the  approximate  value  3y.) 

1.  Prove  that  if  two  oblique  lines  drawn  from  a  point  to  a  straight 
line  meet  this  line  at  unequal  distances  from  the  foot  of  the  per- 
pendicular dropped  upon  it  from  the  given  point,  the  more  remote 
is  the  longer. 

2.  Prove  that  the  distances  of  the  point  of  intersection  of  any 
two  tangents  to  a  circle  from  their  points  of  contact  are  equal. 

A  straight  line  drawn  through  the  centre  of  a  certain  circle  and 


368  PLANE  GEOMETRY 

through  an  external  point,  P,  cuts  the  circumference  at  points 
distant  8  and  18  inches  respectively  from  P.  What  is  the  length 
of  a  tangent  drawn  from  P  to  the  circumference? 

3.  Given  an  arc  of  a  circle,  the  chord  subtended  by  the  arc, 
and  the  tangent  to  the  arc  at  one  extremity,  show  that  the  per- 
pendiculars dropped  from  the  middle  point  of  the  arc  on  the  tangent 
and  chord,  respectively,  are  equal. 

One  extremity  of  the  base  of  a  triangle  is  given  and  the  centre 
of  the  circumscribed  circle.  What  is  the  locus  of  the  middle  point 
of  the  base? 

4.  Prove  that  in  any  triangle  the  square  of  the  side  opposite 
an  acute  angle  is  equal  to  the  sum  of  the  squares  of  the  other  two 
sides  diminished  by  twice  the  product  of  one  of  those  sides  and  the 
projection  of  the  other  upon  that  side. 

Show  very  briefly  how  to  construct  a  triangle  having  given  the 
base,  the  projections  of  the  other  sides  on  the  base,  and  the  pro- 
jection of  the  base  on  one  of  these  sides. 

5.  Show  that  the  areas  of  similar  triangles  are  to  one  another 
as  the  areas  of  their  inscribed  circles. 

The  area  of  a  certain  triangle  the  altitude  of  which  is  -\/2,  is 
bisected  by  a  line  drawn  parallel  to  the  base.  What  is  the  distance 
of  this  line  from  the  vertex? 

6.  Two  flower  beds  have  equal  perimeters.    One  of  the  beds  is 
circular  and  the  other  has  the  form  of  a  regular  hexagon.    The 
circular  bed  is  closely  surrounded  by  a  walk  7  feet  wide  bounded 
by  a  circumference  concentric  with  the  bed.    The  area  of  the  walk 
is  to  that  of  the  bed  as  7  to  9.    Find  the  diameter  of  the  circular 
bed  and  the  area  of  the  hexagonal  bed. 

SEPTEMBER,  1896 

One  question  may  be  omitted 

(In  solving  problems  use  for  ic  the  approximate  value  3y.) 

1.  Prove  that,  if  one  of  two  convex  broken  lines  which  have 
the  same  extremities  envelops  the  other,  the  first  is  the  longer. 

2.  Prove  that,  when  two  circumferences  intersect  each  other, 
the  line  which  joins  their  centres  bisects  at  right  angles  their 
common  chord. 


COLLEGE  ENTRANCE  EXAMINATIONS          369 

The  centres  of  two  circles  of  radii  8  inches  and  6  inches  respec- 
tively are  10  inches  apart.  Show  that  the  common  chord  is  9.6 
inches  long. 

3.  Show  that,  if  two  parallel  tangents  to  a  circle  are  intercepted 
by  a  third  tangent,  the  part  of  the  third  tangent  between  the  other 
two  subtends  a  right  angle  at  the  centre  of  the  circle. 

State  briefly  how  you  might  find  a  fourth  proportional  to  three 
given  straight  lines. 

4.  Prove  that  in  any  obtuse-angled  triangle  the  square  of  the 
side  opposite  the  obtuse  angle  is  equal  to  the  sum  of  the  squares 
of  the  other  two  sides  increased  by  twice  the  product  of  one  of  these 
sides  and  the  projection  of  the  other  upon  that  side. 

What  is  the  locus  of  the  vertices  of  all  the  triangles  so  constructed 
on  a  given  base  that  the  radii  of  their  circumscribed  circles  are  all 
equal  to  a  given  line? 

5.  The  line  passing  through  the  centres  of  two  circles  which 
touch  each  other  externally  at  A,  meets  a  common  tangent,  which 
touches  the  circles  at  B  and  C  respectively,  at  the  point  S.    Show 
that  SA  is  a  mean  proportional  between  SB 

and  SC. 

6.  The  perimeter  of  a  certain  church  win- 
dow is  made  up  of  three  equal  circular  arcs 
the  centres  of  which  are  the  vertices  of  an 
equilateral  triangle.    Each  of  the  arcs  subtends 
an  angle  of  300°  at  its  own  centre.    Find  the 

area  of  the  window,  assuming  the  length  of  the  perimeter  to  be 
110  feet. 

JUNE,  1906 

(ONE  HOUR  AND  A  HALF) 
The  University  Provides  a  Syllabus 

1.  Prove  that  the  sum  of  the  three  angles  of  any  triangle  is 
equal  to  two  right  angles. 

On  a  line  AE  choose  a  point  B,  and  construct  an  isosceles  triangle 
ABC  with  AB  as  base,  the  base  angles  being  less  than  45°.  With 
B  as  vertex  construct  an  isosceles  triangle  BCD  whose  base  CD 
lies  in  AC  produced.  Show  that  the  angle  DBE  is  three  times  the 
angle  A. 


370  PLANE  GEOMETRY 

2.  Prove  that  the  bisector  of  an  angle  of  a  triangle  divides  the 
opposite  side  into  segments  proportional  to  the  sides  of  the  angle. 

The  hypotenuse  of  a  right  triangle  is  10  inches  long  and  one  of 
the  acute  angles  is  30°.  Compute  the  lengths  of  the  segments  into 
which  the  short  side  is  divided  by  the  bisector  of  the  opposite  angle. 

3.  A  chord  BC  of  a  given  circle  is  drawn,  and  a  point  A  moves 
on  the  longer  arc  BC.    Draw  the  triangle  ABC,  and  find  the  locus 
of  the  centre  of  a  circle  inscribed  in  this  triangle. 

4.  Three  equal  circular  plates  are  so  placed 
that  each  touches  the  other  two,  and  a  string 
is  tied  tightly  around  them.     If  the  length 
of  the  string  is  10  feet,  find  the  radius  of  the 
circles  correct  to  three  significant  figures. 

5.  Let  0  be  the  centre  of  a  circle  and  P 
any  point  outside.    With  P  as  centre  and 

radius  PO  draw  the  arc  of  a  circle,  cutting  the  given  circle  at  A  and 
B.  With  A  and  B  as  centres  and  AB  as  radius  draw  arcs  inter- 
secting in  Q.  Prove: 

(a)  that  the  points  0,  Q,  P  are  in  a  straight  line;  and 

(6)  thatOPxOQ  =  (OA)\ 

Suggestion. — Join  A  and  B  with  0,  P,  and  Q,  and  join  Q  with 
0  and  P. 

SEPTEMBER,  1907 

(ONE  HOUR  AND  A  HALF) 

The  University  Provides  a  Syllabus 

1.  Prove  that  in  an  isosceles  triangle  the  angles  opposite  the 
equal  sides  are  equal. 

On  BC,  the  longest  side  of  a  triangle  ABC,  points  B'  and  C"  are 
taken  so  that  BB'^-BA  and  CCf  =  CA.  Show  that  the  angle 
B'AC'  equals  half  the  sum  of  the  angles  B  and  C. 

2.  Prove  that  the  area  of  a  triangle  is  one-half  the  product  of 
the  base  and  the  altitude. 

Show  that  if  a  point  move  about  within  a  regular  polygon,  the 
sum  of  the  perpendiculars  let  fall  upon  the  sides  (or  the  sides 
produced)  will  be  constant. 


COLLEGE  ENTRANCE  EXAMINATIONS          371 

3.  A  rod  8  feet  long  is  free  to  move  within  a  rectangle  8  feet 
long  and  6  feet  wide.     Describe  accurately   the    boundary    of 
the  region  within  which  the  middle  point  of  the  rod  will  always 
be  found. 

4.  A    circle    is    described    upon   one    side   of   an    equilateral 
triangle   as   diameter.      Compute  the  area   of  the  part  of  the 
triangle  which  lies  outside  the  circle,  correct  to  one  per  cent,  of  its 
value. 

5.  Two  circles  intersect  at  right  angles.     The  radius  of  one  of 
them  is  of  length  a,  and  its  centre  is  the  point  0.    Show  that  if 
any  line  be  drawn  through  0  cutting  the  second  circle  in  the  points 
P  and  P',  then 


1908 

(ONE  HOUR  AND  A  HALF) 
The  University  Provides  a  Syllabus 

1.  Prove  that  if  in  a  quadrilateral  a  pair  of  opposite  sides  be 
equal  and  parallel,  the  figure  is  a  parallelogram. 

In  a  certain  quadrilateral,  one  diagonal,  and  a  line  connecting 
the  middle  points  of  a  pair  of  opposite  sides  bisect  each  other. 
Prove  that  the  quadrilateral  is  a  parallelogram. 

2.  Prove  that  in  any  circle  equal  chords  subtend  equal  arcs. 
Show  that  the  bisector  of  an  angle  of  a  triangle  meets  the  per- 

pendicular bisector  of  the  opposite  side  on  the  circumference  of  the 
circumscribed  circle. 

3.  The  radii  of  two  circles  are  1  inch,  and  V  3  inches  respectively, 
and  the  distance  between  their  centres  is  2  inches.    Compute  their 
common  area  to  three  significant  figures. 

4.  Determine  a  point  P  without  a  given  circle  so  that  the  sum  of 
the  lengths  of  the  tangents  from  P  to  the  circle  shall  be  equal  to 
the  distance  from  P  to  the  farthest  point  of  the  circle. 

5.  The  image  of  a  point  in  a  mirror  is,  apparently,  as  far  behind 
the  mirror  as  the  point  itself  is  in  front.     If  a  mirror  revolve  about 
a  vertical  axis,  what  will  be  the  locus  of  the  apparent  image  of  a 
fixed  point  one  foot  from  the  axis? 


372  PLANE  GEOMETRY 

1909 

(Two  HOURS) 

The  University  Provides  a  Syllabus 

1.  Show  that  if  a  triangle  be  equilateral,  it  is  also  equiangular. 
Is  this  theorem  true  in  the  case  of  a  quadrilateral?    Give  your 

reason. 

2.  Prove  that  the  square  of  the  hypotenuse  of  a  right  triangle  is 
equal  to  the  sum  of  the  squares  of  the  other  two  sides. 

Deduce  from  this  a  proof  that  of  two  oblique  lines  drawn  from 
the  same  point  of  a  perpendicular  and  cutting  off  unequal  distances 
from  the  foot  of  the  perpendicular,  the  more  remote  is  the  greater. 

3.  Prove  that  if  the  products  of  the  segments  into  which  the 
diagonals  of  a  quadrilateral  divide  one  another  are  equal,  a  circle 
may  be  circumscribed  about  the  quadrilateral. 

4.  A  square  10  inches  on  a  side  is  changed  into  a  regular  octagon 
by  cutting  off  the  corners.    Find  the  area  of  this  octagon. 

5.  A  wheel  40  inches  in  diameter  has  a  flat  place  5  inches  long 
on  the  rim.    Describe  carefully  the  locus  of  the  centre  as  the  wheel 
rolls  along  the  level. 

1910 

(Two  HOURS) 
The  University  Provides  a  Syllabus 

1 .  Prove  that  if  the  sides  of  one  angle  be  perpendicular  respectively 
to  those  of  another,  the  angles  are  either  equal  or  supplementary. 

2.  Show  how  to  inscribe  a  circle  in  a  given  triangle. 

How  many  circles  can  be  drawn  to  touch  three  given  lines?  Are 
there  any  positions  of  these  lines  for  which  the  number  is  less? 

3.  Define ''incommensurable  magnitudes."    Give  a  careful  proof 
of  some  theorem  where  such  magnitudes  occur. 

d.  A  BCD  are  the  vertices  in  order  of  a  quadrilateral  which  is 
circumscribed  to  a  circle  whose  centre  is  0.  Prove  that  /^AOB 
and  /_  COD  are  supplementary. 

5.  Two  radii  of  a  circle  OA  and  OB  make  a  right  angle.  A  second 
circle  is  described  upon  A  B  as  diameter.  Prove  that  the  area  of  the 
crescent-shaped  region  outside  of  the  first  circle,  but  inside  of  the 
second,  is  equal  to  that  of  the  triangle  AOB.  (Hippocrates,  fifth 
century  B.C.) 


COLLEGE  ENTRANCE  EXAMINATIONS         373 

1915 

(Two  HOURS) 
The  University  Provides  a  Syllabus 

1.  Two  straight  lines  are  cut  by  a  third,  and  the  alternate 
interior  angles  are  equal.     Prove  that  the  two  straight  lines  are 
parallel. 

Prove  that  the  bisector  of  an  exterior  angle  at  the  vertex  of  an 
isosceles  triangle  is  parallel  to  the  base. 

2.  Prove  that  an  angle  formed  by  a  tangent  to  a  circle  and  a 
chord  through  the  point  of  contact  is  measured  by  one-half  of 
the  intercepted  arc. 

Tangents  are  drawn  at  the  extremities  of  a  chord  of  a  circle,  and 
the  perpendicular  bisector  of  the  chord  is  drawn.  The  point  in 
which  this  last  line  cuts  the  minor  arc  is  connected  with  one  end 
of  the  chord.  Show  that  this  connecting  line  bisects  the  angle 
between  the  chord  and  the  tangent. 

3.  Two  unequal  circles  are  tangent  to  each  other  externally,  and 
their  common  tangent  at  the  point  of  contact  meets  one  of  the  other 
common  tangents  at  the  point  P.    Lines  are  drawn  from  P  to  the 
centres  of  the  two  circles.    Prove  that  these  X*"^X 

two  lines  are  perpendicular.  /  \ 

4.  Three  equal  circular  plates  of  radius  r       /  \  J  \ 
are  so  placed  that  each  is  tangent  to  the  other    /"     ^\~/~  "\ 
two.      Find  the  length  of  the  shortest  string  f               Y  i 
that  can  be  tied  around  the  three  circles,  and  V              A             J 
the  area  enclosed  by  this  string.                         ^-    <S  ^- — S 

5.  Two  circles  ABC  and  ADE  touch  internally  at  A,  and  through 
A  straight  lines,  ABD  and  ACE,  are  drawn  to  cut  the  circles. 
Prove  that  AB-DE  =  AD-BC. 


CHAPTER  X 

SUGGESTIONS 

The  aim  of  this  chapter  is  to  suggest  appropriate  material  for 
reading  and  discussion  outside  of  class,  and  in  mathematics  clubs. 
To  this  end  certain  books  and  topics  are  mentioned,  but  it  is 
understood  that  these  lists  are  intended  to  be  suggestive  rather 
than  comprehensive,  and  include  only  subject  matter  adapted  to 
pupils  at  the  stage  of  development  represented  by  this  text. 

After  topics  in  the  second  list,  parenthetical  reference  is  fre- 
quently made  to  books  from  the  first  list,  in  which  at  least  some- 
thing about  the  topic  may  be  found,  and  in  the  third  list  definite 
references  for  a  few  topics  are  given  as  a  guide  to  the  beginner. 
The  practice  of  seeking  such  references  without  help  is  particularly 
valuable  as  a  training  for  college  work;  hence,  this  list  includes 
only  a  few  topics,  and  is  to  serve  for  direction  at  the  outset  only. 

A.  LIST  OF  TOPICS  SUITABLE  FOR  STUDENTS'  DISCUSSION 

GENERAL 

Mathematical  Games.    (Ball.) 

Card  Tricks. 

Problems  on  a  Chess  Board. 

History  and  Elementary  Idea  of  Calculus.    (White.) 
Some  Applications  of  Mathematics  to  Astronomy.    (Ball.) 
Parcel  Post  Problems.    (School  Science  and  Mathematics.) 
Mathematics  of  Common  Things. 

Optical  Illusions.    (Smith's  Teaching  of  Geometry.) 
Navigation.    (Richards.) 
Instruments. 

Historic.     (School  Science  and  Mathematics.) 

Astrolabe,  squadra,  carpenter's  level,  baculus  mensorus,  sundial,  etc. 
Pantograph.     (Phillips  and  Fisher's  Elements  of  Geometry.) 

Map-making. 

Planimeter.     (Scientific  American.) 
Symmetry.    (White;  Dobbs,  Symmetry,  Chapt.  VI.) 

In  nature. 

Design. 
Maxima  and  Minima.    (Texts  on  Plane  Geometry.) 

374 


SUGGESTIONS  375 

Linkages.    (White.) 

Fourth  Dimension.    (Flatland.) 

Angles  of  a  General  Polygon.    (School  Science  and  Mathematics.) 

Study  of  Rupert's  "Famous  Problems  in  Geometry."    (Heath's  Monographs.) 

Fallacies.    (Ball.) 

Odd  or  Purely  Mechanical  Constructions.    (Becker.) 

ARITHMETIC 

History  of.     (Ball,  Fink,  Cajori,  Brooks.) 

Primitive  Numeration. 

Systems  of  Notation. 

Problems  in  Number  Systems  Other  Than  Our  Own. 

Special  Consideration  of  Duodecimal  System. 

Kinds  of  Number. 

Fundamental  Operations. 

Russian  Peasant  Multiplication. 
Calculation.    (Langley's  Computation.    Longmans,  Green  &  Co.) 

Short  Cuts.    (Jones;  White.) 

Calculating  Machines. 

Slide  Rule.    (The  Teaching  of  Mathematics,  by  Schultze,  Macmillan.) 

Approximation.    (White.) 

Repeating  Decimals.    (White.) 

Peculiarities  and  Possibilities  of  Our  Electoral  System. 
Number  Curiosities. 

9  and  its  properties.    (White.) 

Tricks.    (Dudeney;  Jones;  White.) 
Alligation.    (The  Mathematics  Teacher.) 

ALGEBRAIC 

History  of.     (Ball,  Fink,  Cajori,  etc.) 

Stages — Rhetorical,  Syncopated  and  Symbolic. 

Symbols. 

Fallacies.    (White;  Jones.) 
Choice  and  Chance.    (Text  on  higher  algebra.) 

GEOMETRIC 

History.    (Smith's  Teaching  of  Geometry.) 

More  detailed  study  of  some  period  other  than  is  given  in  text. 
Famous  Problems.    (Heath's  Monographs.) 
Trisection  of  an  Angle. 

Instruments. 
Squaring  of  the  Circle. 
Pythagorean  Proposition. 

Various  proofs  and  applications. 


376  PLANE  GEOMETRY 

B.  TOPICS  WITH  DEFINITE  REFERENCES 

GEOMETRIC  FALLACIES 

Right  angle  is  obtuse.    Ball's  Recreations,  p.  40. 

Part  of  a  sect  equals  the  sect.    Ball's  Recreations,  p.  41. 

Every  triangle  is  isosceles.    Ball's  Recreations,  p.  42. 9 

Miscellaneous  fallacies.    Ball's  Recreations,  pp.  43-46. 

Hypotenuse  equals  sum  of  legs  of  triangle.    Canterbury  Puzzles,  p.  26. 

1  =0.    Wrinkles,  p.  93. 

Two  perpendiculars  from  a  point  to  a  line.    Wrinkles,  p.  95. 

NUMBER  CURIOSITIES 

X 

Mystic  Properties  of  Numbers. 

Repeating  products.    White,  p.  11  et  seq. 

Nine.    White,  p.  25. 
Fallacies.    Ball,  p.  23  et  seq. 
Magic  Squares.    See  Andrews's  book  and  his  references. 

PYTHAGOREAN  PROPOSITION 

Various  proofs. 

Rupert's  "Famous  Geometrical  Theorems  and  Problems." 
Applications  (curious). 

Fly  Problem.    Jones  p.  2,  No.  10. 

Historical  Problems.    See  Beman  and  Smith's  Academic  Algebra,  p.  153. 

C.  LIST  OF  BOOKS  SUITABLE  FOR  STUDENTS'  READING. 

HISTORY 

1  ALLMAN:  Greek  History  from  Thales  to  Euclid.    Longmans. 
BALL:  History  of  Mathematics.     Macmillan. 

Primer  of  the  History  of  Mathematics.    Macmillan. 
BROOKS:  Philosophy  of  Arithmetic.    Normal  Publishing  Co. 
CAJORI:  History  of  Mathematics.      Macmillan. 

History  of  Elementary  Mathematics.    Macmillan. 
CON  ANT:  Number  Concept.    Macmillan. 
FINE:  Number  Systems  of  Algebra.    Heath. 
FINK:  Brief  History  of  Mathematics.    Open  Court  Pub.  Co. 
FRANKLAND:  The  Story  of  Euclid.    Wessel  and  Co. 
Gow:  History  of  Greek  Mathematics.    Cambridge  Press. 
KLEIN:   Famous  Problems  of  Elementary  Geometry.    Ginn. 
HEATH:   Diophantus  of  Alexandria.    Cambridge  Press. 
MANNING:  Non-Euclidean  Geometry.    Ginn. 

MILLER,   G.   A.:     Historical   Introduction    to    Mathematical    Literature. 
Macmillan. 


SUGGESTIONS  377 

SMITH,  D.  E. :  Kara  Arithmetica.    Ginn. 

Teaching  of  Geometry.    Ginn. 

Teaching  of  Elementary  Mathematics.    Macmillan. 
SMITH  AND  KARPINSKI:     The  Hindu-Arabic  Numerals.     Ginn. 

History  of  Japanese  Mathematics.    Open  Court  Pub.  Co. 
BOYBR:   Histoire  des  Mathematiques.     Paris. 
CANTOR:   Vorlesungen  Uber  Geschichte  der  Mathematik.    Leipzig. 
Portraits  of  Mathematicians  by  Prof.  D.  E.  SMITH. 

Portfolios.    Open  Court  Pub.  Co. 
Lantern  Slides  by  Prof.  D.  E.  SMITH,  of  Teachers'  College,  Columbia  University. 

RECREATIONS 

ABBOTT:  Flatland.    Little,  Brown. 

ANDREWS:  Magic  Squares  and  Cubes.    Open  Court  Pub.  Co. 
ANONYMOUS:  Flatland.    Boston. 
BALL:  Mathematical  Recreations.    Macmillan. 
CAVENDISH:  Recreations  with  Magic  Squares.    London. 
DUDENEY:  The  Canterbury  Puzzles.    Button  &  Co. 
HARPSON:   Paradoxes  of  Nature  and  Science.    Dutton  &  Co. 
HATTON:  Recreations  in  Mathematics.    London. 
HILL:  Geometry  and  Faith.    Lee  and  Shepard. 
JONES,  S.  I. :  Mathematical  Wrinkles.    Author,  Gunther,  Texas. 
KEMPE:  How  to  Draw  a  Straight  Line.    Macmillan. 
LATOON:  On  Common  and  Perfect  Magic  Squares.    Cambridge. 
DE  MORGAN:  A  Budget  of  Paradoxes.    London. 
MANNING:  Fourth  Dimension  Simply  Explained.    Munn. 
PERRY:  Spinning  Tops.    London. 

SCHOFIELD:  Another  World.    Swan,  Sonnenschein;  London. 
SCHUBERT:  Mathematical  Recreations.    Open  Court  Pub.  Co. 
WHITE:  Scrap  Book  of  Elementary  Mathematics.    Open  Court  Pub.  Co. 
AHRENS:   Mathematische  Unterhaltungen  und  Spiele.    Leipzig. 
LUCAS:  Recreations  Mathematiques.    Paris. 
L'Arithmetique   amusante.    Paris. 

MAUPIN:   Opinions  et  curiosites  touchant  la  mathe"matique.    Paris. 
REBIERE:    Mathematiques  et  Mathematiciens,  Pensee"s  et  Curiosites.    Paris. 

PRACTICAL 

BRECKENRIDGE,  MERSEREAU  AND  MOORE:    Shop  Problems  in  Mathematics. 

Ginn. 

CALVIN,  F.  H.:  Shop  Calculations.    McGraw  Hill. 
CASTLE:   Manual  of  Practical  Mathematics.    Macmillan. 
COBB:   Applied  Mathematics.    Ginn. 
Cox:    Manual  of  Slide  Rule.    Keuffel  and  Esser. 
DOOLEY:    Vocational  Mathematics.     Heath. 
HINDS,  NOBLE  AND  ELDREDGE:    How  to  Become  Quick  at  Figures. 


378  PLANE  GEOMETRY 

LANGLEY:  Computation.    Longmans. 

MARSH  :  Vocational  Mathematics.    Wiley  and  Sons. 

MURRAY:  Practical  Mathematics.    Longmans. 

RICHARDS:  Navigation  and  Nautical  Astronomy.    American  Book  Co. 

SAXELBY:  Practical  Mathematics.    Longmans. 

GENERAL 

CARUS:  Foundations  of  Mathematics.    Open  Court  Pub.  Co. 

CLIFFORD  :  Common  Sense  of  the  Exact  Sciences.    Appleton. 

FRANKLAND:  Theories  of  Parallelism.    Cambridge  Press. 

HENRICI:  Congruent  Figures.    Longmans. 

LAGRANGE:  Lectures  on  Elementary  Mathematics.    Open  Court  Pub.  Co. 

Row:  Geometric  Paper  Folding.    Open  Court  Pub.  Co. 

SYKES:  Source  Book  of  JProblems  for  Geometry.    Allyn  and  Bacon. 

VON  H.  BECKER:  Geometrisches  Zeichnen.    Sammlung  Goechen. 


INDEX 


Abbreviations,  xi 
Abscissa,  149 
Acute  angle,  28 

triangle,  37 

Addition,  in  proportion,  107 
Adjacent,  31 
Ahmes,  5 
Alternate,  exterior,  53 

interior,  53 
Alternation,  107 
Altitude,  of  parallelogram,  80 

of  rectangle,  78 

of  trapezoid,  82 

of  triangle,  81 
Analysis,\  297 

of  problems,  319 
Angle,  acute,  28 

adjacent,  31 

bisection,  19 

central,  161 

central,  of  regular  polygon,  181 

complementary.  28 

construction,  19 

definition  of,  27 

exterior,  of  polygon,  60 

inscribed,  174 

measurement,  28,  31 

obtuse,  28 

of  depression,  142 

of  elevation,  142 

right,  28 

straight,  28 

subtended  by  a  circle,  284 

supplementary,  28 

vertical,  32,  37 
Angles,  alternate,  53 

consecutive,  53 

exterior,  53 

interior,  53 
Antecedent,  105 


Antilogarithm,  95 
Apollonius,  problem,  358 
Apothem,  186 
Arc,  161 

major,  161 

minor,  161 
Arch,  Ogee,  170 

Persian,  170 
Archytas,  8 
Area,  75 

of  circle,  180 

of  parallelogram,  81 

of  rectangle,  77 

of  trapezoid,  83 

of  triangle,  81 
Axioms,  25,  49 

of  inequality,  227,  273 

of  variables,  187 

Base,  in  logarithms,  86 

of  rectangle,  78 

of  trapezoid,  82 

of  triangle,  81 
Bisection,  of  angle,  19 

of  line,  18 

Bibliography,  historical,  10 
Boyle's  Law,  109 

Cam,  23 
Cancellation,  74 
Center,  of  circle,  160 

of  gravity,  72 

of  parallelograms,  69 

of  similitude,  119 
Central  angle,  161 

of  regular  polygon,  181 
Centroid,  359,  360 
Ceradini's  Method,  294 
Characteristic,  92 
Charles's  Law,  110 


379 


380 


INDEX 


Chord,  162 

common,  169 
Chord  of  contact,  281 
Circle,  definition  of,  160 

equation  of,  155 
Circumcenter,  307 
Circumference,  160 
Circumscribed  circle,  180 

n-gon,  180 
Commensurable,  77 
Common  chord,  169 

measure,  how  to  find,  307 
Complement,  28 
Composition,  107 
Concurrent,  267 
Congruence,  definition  of,  38 
Consecutive,  exterior,  53 

interior,  53 
Consequent,  105 
Constant,  184 
Contact,  chord  of,  281 
Continuity,  principle,  261 
Contradictory,  of  the  direct,  300 
Converse,  56,  57 

theorem,  155 
Converses,  law  of,  300 
Coordinate,  149 
Coplanar,  51 
Corollary,  30 
Corresponding  angles,  53 
Cosine,  139 
Curve,  reversed,  169 

Decagon,  63 

construction  of  regular,  316 
Definitions,  24 
Denominator,  74 
Depression,  angle  of,  142 
Designs,  20,  21,  22 
Diagonal,  62 

scale,  127 
Diameter,  160 
Dimension  of  rectangle,  78 
Direct  method,  290 


Direct  theorem,  155 

variation,  110 
Discussion  of  solution,  311 
Distance,  48 

between  parallels,  67 
Dividing  a  sect,  117 
Division,  108 

external,  244 

harmonic,  244 

internal,  244 
"Doubling  the  angle  at  the  bow,"  67 

Elevation,  angle  of,  142 
Elimination,  228 
Equation,  as  locus,  152 

of  circle,  155 
Equilateral  triangle,  37 
Escribed  circle,  314 
Euclid,  9 
Eudoxus,  9 
Excenter,  314 
Exclusion,  228,  297 
Exponents,  fractional,  88 

negative,  88 

principles  of,  87 

zero,  88 

Exterior  angle,  63 
External  division,  244 

tangent  circles,  168 
Extreme  and  mean  ratio,  315 
Extremes.  106 

Foot  of  perpendicular  to  a  line,  134 
Formulas,  summary,  336,  et  seq. 
Fourth  proportional,  construction,  308 
Fraction,  74 

Geometry,  derivation,  9 
Golden  Section,  315 
Graph,  150 

application  of,  150 

of  equations,  151 
Gravitation,  Law  of,  111,  112 


INDEX 


381 


Harmonic  division,  244,  307 

Heptagon,  63 

Heron  of  Alexandria's  formula,  98, 261 

Hexagon,  63 

Hexagram,  Mystic,  288 

Hippocrates'  Theorem,  296 

Homologous,  definition,  42 

Hypotenuse,  50 

Hypothesis,  54 

Hypsometer,  126 

Incenter,  314 

Inclined  plane,  law  of,  109 
Incommensurable,  77 
Indirect  method,  297 
Inequality  axioms,  227,  273 
Initial  side,  139 
Inscribed  angle,  174 

circle,  180 

n-gon,  180 
Intercept,  162 
Interest,  compound,  98 
Internal  division,  244 

tangent  circle,  168 
Interpolation,  94 
Intersection  of  loci,  312 
Inverse  variation,  110 
Inversion,  108 
Isosceles  triangle,  37 

Joint  variation,  111 

Law  of  converses,  300 
Limits,  184 

inferior,  184 

postulate  of,  184 

superior,  184 
Line,  14 

broken,  228 

curved,  15 

of  centers,  168 

straight,  15 
Lines,  parallel,  51 
Locus,  152,  153,  269,  312 


Logarithm,  86 

tables,  103 
Logarithms,  common,  91 

historical  note,  90 

theorems,  93 

Major  arc,  161 

Mantissa,  92 

Mean  and  extreme  ratio,  315 

proportion,  106 

proportional,  106 

construction,  315 
Means,  106 
Measure,  173 
Median,  230 

of  trapezoid,  244 
Method,  direct,  297 

indirect,  297 

synthetic,  297 
Methods    of   proof,   summary,    338, 

et  seq. 
Minor  arc,  161 

Nonagon,  63 
Numerator,  74 

Obtuse  angle,  28 

triangle,  37 
Octagon,  63 
Opposite  theorem,  155 
Ordinate,  149 
Origin,  149 
Orthocenter,  359,  360 

Pantograph,  131 
Parallel,  51 

postulate,  52 
Parallelogram,  67 

altitude  of,  80 

base  of,  80 

of  forces,  71 

ruler,  68,  72 

Parallels,  construction,  55 
TT,  188 


382 


INDEX 


TT,  historical  note,  188 
Pentagon,  63 

construction  of  regular,  317 
Pentagram,  234 
Penumbra,  179 

Perpendicular,  construction,  17,  18 
Plane,  15 
Plato,  8 
Plumb-level,  52 
Point,  14 
Polygon  circumscribed  about  a  circle, 

180 
Polygon,  convex,  63 

definition,  37 

inscribed  in  a  circle,  180 

regular,  63 

similar,  construction,  308 
Pons  asinorum,  46 
Postulate,  26,  27 

of  angles,  31 

of  limits,  184 

of  parallels,  52 

superposition,  38 

Postulates  of  perpendiculars,  48,  161 
Projection,  133 

orthogonal,  135 

point  upon  a  line,  134 

sect  upon  a  line,  134 
Projector,  139 
Proportion,  106 
Proportional  compasses  or  dividers, 

129 

Protractor,  29 
Pythagoras,  7 
Pythagorean  badge,  234 

numbers,  12 

theorem,  135 

Quadrant,  35 
Quadrilateral,  63 

Radian,  191 
Radius,  160 
of  regular  polygon,  186 


Ratio,  74 

extreme  and  mean,  315 
Ratio  of  similitude,  119 
Rectangle,  69 

altitude,  78 

area,  80 

base,  78 

dimensions,  78 

"Reductio  ad  absurdum,"  300 
Reduction  to  an  absurdity,  297 
Regular  polygon,  63 
Resultant,  70 
Rhomboid,  238 
Right  angle,  28 

Scalene  triangle,  37 
Secant,  173 
Sect,  15 
Sector,  192 

compasses,  129 
Sectors,  similar,  291 
Segment,  291 
Segments,  similar,  291 
Sextant,  62 
Similar,  119 

figures,  119 

polygon,  construction,  308 

sectors,  291 

segments,  291 

Similarity  of  sets  of  points,  119 
Similitude,  center  of,  119 

ratio  of,  119 
Simpson's  Rule,  84 
Sine,  139 
Slide  rule,  98 
Solid,  14 

Specific  gravity,  109 
Speculum,  125 
Squadra,  135 
Square,  carpenters',  175 
" Square  of,"  239 
" Square  on,"  239 

optical,  62 

root  scale,  138 


INDEX 


383 


Star  polygon,  234 
Straight  angle,  28 
Subtraction,  in  proportion,  108 
Summary  of  formulas,  336  et  seq. 
of  methods  of  proof,  338  et  seq. 
Superposition,  postulate,  38 
Supplement,  28 
Surface,  14 
Symbols,  xi 
Synthetic  method,  297 

Tangent,  139,  178 

circles,  168 
external,  168 
internal,  168 

common,  construction,  314 

construction,  313 

line,  165 

Terminal  side,  139 
Terms  of  fraction,  74 

of  proportion,  106 
Thales,  7 
Theodolite,  35 
Theorem,  36 
Third    proportional,    construction, 

308 

Transforming  a  polygon,  317 
Transversal,  53 
Trapezoid,  altitude,  82 

area,  82 

bases,  82 


Trapezoid,  definition,  82 

Egyptian  formula  for  area,  11 

median,  244 
Trapezoidal  rule,  84 
Trefoil,  170,  171 
Triangle,  37 

acute,  37 

altitude,  81 

area,  81 

base,  81 

equilateral,  37 

isosceles,  37 

medians  of,  230 

obtuse,  37 

right,  37   « 

scalene,  37 
Trigonometric  ratios,  139 

Umbra,  179 

Variable,  184 

decreasing,  184 

increasing,  184 
Variation,  110 

direct,  110 

inverse,  110 

joint,  111 
Vertex  of  angle,  27 

of  polygon,  37 
Vertical  angle,  37 


1C  49575 


•T  f 


IVERSITY  OF  CALIFORNIA  LIBRARY  . 


